3.3: Functions and Function Notation
- Determine whether a relation represents a function.
- Evaluate functions.
- Find domain of functions.
Try these questions prior to beginning this section to help determine if you are set up for success:
- Evaluate the expression at the given value: \[x^2-8x-13 \text{ at } x=-1\nonumber\]
-
Simplify:
- \(\quad -4-(b-1)^2\)
- \(\quad\dfrac{3}{3-m}-\dfrac{2m}{m+3}+\dfrac{12-8m}{m^2-9}\)
- \(\quad\dfrac{1-\frac{4}{t^2}}{1-\frac{2}{t}}\)
- Solve: \[2y^3-3y^2=8y-12\nonumber\]
- Solve the inequality and write your answer in interval notation: \[x(6x+1) \geq 12 \nonumber\]
- Answer
-
- \(-4\)
If you missed this problem or feel you could use more practice, review [ 2.3: Polynomial Expressions ]
-
- \(\quad -b^2+2b-5\). If you missed this problem or feel you could use more practice, review [ 2.6: Multiplying Polynomial Expressions ]
- \(\quad -\dfrac{2m-1}{m-3} \nonumber\). If you missed this problem or feel you could use more practice, review [ 2.14: Adding and Subtracting Rational Expressions ]
- \(\quad\dfrac{t+2}{t} \nonumber\). If you missed this problem or feel you could use more practice, review [ 2.15: Simplifying Complex Rational Expressions ]
- \(y=-2 \text{, } 0 \text{, } \dfrac{3}{2} \)
If you missed this problem or feel you could use more practice, review [ 2.8: Solving Polynomial Equations and 2.12: General Strategy for Factoring Polynomials ]
- \(\left(-\infty,-\dfrac{3}{2}\right) \cup \left(-\dfrac{3}{2},\dfrac{4}{3}\right) \cup \left(\dfrac{4}{3},\infty\right)\)
If you missed this problem or feel you could use more practice, review [ 3.2: Solving Nonlinear Inequalities ]
Imagine visiting a faraway city or even outer space from the comfort of your living room. It could be possible using virtual reality. This technology creates realistic images that make you feel as if you are truly immersed in the scene and even enable you to interact with them. It is being developed for fun applications, such as video games, but also for architects to plan buildings, car companies to design prototypes, the military to train, and medical students to learn. Developing virtual reality devices requires modeling the environment using functions.
What is a Function?
The natural world is full of relationships between quantities that change. When we see these relationships, it is natural for us to ask "If I know one quantity, can I then determine the other?" This establishes the idea of an input quantity, or independent variable, and a corresponding output quantity, or dependent variable. From this we get the notion of a functional relationship in which the output can be determined from the input.
For some quantities, like height and age, there are certainly relationships between these quantities. Given a specific person and any age, it is easy enough to determine their height, but if we tried to reverse that relationship and determine age from a given height, that would be problematic, since most people maintain the same height for many years.
Determining Whether a Relation Represents a Function
A relation is a set of ordered pairs. The set of the first components of the ordered pairs is called the domain and the set of the second components of the ordered pairs is called the range .
Consider the following set of ordered pairs: \(\{(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)\}\)
The domain is \(\{1, 2, 3, 4, 5\}\). The range is \(\{2, 4, 6, 8, 10\}\).
Each value in the domain is also known as an input value. Each value in the range is also known as an output value.
A function is a relation that assigns each value in the domain to a single value in the range.
To determine if a relation represents a function we can use a mapping diagram .
Figure \(\PageIndex{1}\) shows mapping diagrams for relations that are functions and are not functions.
A function is a relation in which each input value in the domain is assigned to exactly one output value in the range. We say “the output is a function of the input.”
The set of all input values are called the domain values, and the set of all output values are called the range values.
The coffee shop menu, shown in Figure \(\PageIndex{2}\) consists of items and their prices.
- Is price a function of the item?
- Is the item a function of the price?
Solution
- Let’s begin by considering the input as the items on the menu. The output values are then the prices. See Figure \(\PageIndex{3}\).
Each item on the menu has only one price, so the price is a function of the item.
- Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See Figure \(\PageIndex{4}\).
Therefore, the item is a not a function of price.
In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? Table \(\PageIndex{1}\) shows a possible rule for assigning grade points.
| Percent grade | 0–56 | 57–61 | 62–66 | 67–71 | 72–77 | 78–86 | 87–91 | 92–100 |
| Grade point average | 0.0 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 |
Solution
For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.
In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.
Table \(\PageIndex{2}\) lists the five greatest baseball players of all time in order of rank.
| Player | Rank |
|---|---|
| Babe Ruth | 1 |
| Willie Mays | 2 |
| Ty Cobb | 3 |
| Walter Johnson | 4 |
| Hank Aaron | 5 |
- Is the rank a function of the player name?
- Is the player name a function of the rank?
- Answer a
-
Yes
- Answer b
-
Yes. (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)
Function Notation
Once we determine that a relation is a function of the input, we can define a name to describe this relationship. This is especially useful in programming. We typically use a letter or symbol when naming a function. The letters \(f\), \(g\),and \(h\) are often used to represent functions. Sometimes we refer to the function relationship as a rule.
For example, to represent “height is a function of age,” we start by identifying variables for the domain and range. Let \(a\) represent age and \(h\) represent height. Next, define a name for the function. Let \(f\) be the function name.
\[\begin{array}{ll} h \text{ is } f \text{ of }a \;\;\;\;\;\; & \text{We name the function }f \text{; height is a function of age.} \\ h=f(a) & \text{We use parentheses to indicate the function input.} \\ f(a) & \text{We name the function }f \text{ ; the expression is read as “ }f \text{ of }a \text{.”}\end{array}\nonumber\]
The notation \(f(a)\) shows us that \(h\) depends on \(a\). The value \(a\) must be put into the function \(f\) to get an output \(h\). The parentheses indicate that age is the input for the function; they do not indicate multiplication.
Note that the input and output of a function do not have to be numbers. They can be names of people, labels of geometric objects, etc. Most of the functions we will work with in this book will have numbers as inputs and outputs.
For the function \(y=f(x)\)
\[\begin{array} {l} {f\text{ is the name of the function}} \\{x \text{ is the domain value}} \\ {f(x) \text{ is the range value } y \text{ corresponding to the value } x} \\ \nonumber \end{array}\]
We read \(f(x)\) as \(f\) of \(x\) or the value of \(f\) at \(x\).
We call x the independent variable as it can be any value in the domain. We call y the dependent variable as its value depends on x .
Use function notation to represent a function whose input is the name of a month and output is the number of days in that month.
Solution
The number of days in a month is a function of the name of the month, so if we name the function \(f\), we write \(\text{days}=f(\text{month})\) or \(d=f(m)\). The name of the month is the input to a “rule” that associates a specific number (the output) with each input.
For example, \(f(\text{March})=31\), because March has 31 days. The notation \(d=f(m)\) reminds us that the number of days, \(d\) (the output), is dependent on the name of the month, \(m\) (the input).
Analysis
Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.
A function \(N=f(y)\) gives the number of police officers, \(N\), in a town in year \(y\). What does \(f(2005)=300\) represent?
Solution
When we read \(f(2005)=300\), we see that the input year is 2005. The value for the output, the number of police officers \((N)\), is 300. Remember, \(N=f(y)\). The statement \(f(2005)=300\) tells us that in the year 2005 there were 300 police officers in the town.
Instead of a notation such as \(y=f(x)\), could we use the same symbol for the output as for the function, such as \(y=y(x)\), meaning “\(y\) is a function of \(x\)?”
Yes, this is often done with formulas, especially in applied subjects that use higher level math, such as physics and engineering. While exploring math itself we like to maintain a distinction between a function such as \(f\) , which is a rule or procedure, and the output y we get by applying \(f\) to a particular input \(x\) . This is why we usually use notation such as \(y=f(x),P=W(d)\) , and so on.
Evaluating a Function
When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.
Let's look at the function \(y=4x−5\). To find the value of y when \(x=2\), we know to substitute \(x=2\) into the function and then simplify.
| \(y=4x-5\) | |
| Let x=2. | \(y=4\cdot 2-5\) |
| \(y = 3 \) |
The value of the function at \(x=2\) is 3.
We do the same thing using function notation, the equation \(y=4x−5\) can be written as \(f(x)=4x−5\). To find the value when \(x=2\), we write:
| \(f(x) = 4x-5\) | |
| Let x=2. | \(f(2)=4 \cdot {\color{Red}2} -5 \) |
| \(f(2) = 3 \) |
The value of the function at \(x=2\) is 3.
This process of finding the value of \(f(x)\) for a given value of x is called evaluating a function .
For the function \(f(x)=2x^2+3x−1\), evaluate the function.
- \(f(3)\)
- \(f(−2)\)
- \(f(a)\)
Solution
a.
| \(f(x)=2 x^2+3 x-1\) | |
| To evaluate \(f(3)\), substitute 3 for x . | \(f(3)=2(3)^2+3 \cdot 3-1\) |
| Simplify. | \(f(3)=2 \cdot 9+3 \cdot 3-1\) |
| \(f(3)=18+9-1\) | |
| \(f(3)=26\) |
b.
| \(f(x) = 2\) | ||
| To evaluate \(f(-2)\), substitute \(-2\) for \(x\). | \(f(-2)=2(-2)^2+3(-2)-1\) | |
| Simplify. | \(f(-2)=2 \cdot 4+(-6)-1\) | |
| \(f(-2)=8+(-6)-1\) | ||
| \(f(-2)=1\) |
c.
| \(f(x)=2 x^2+3 x-1\) | |
| To evaluate f(a),f(a), substitute a for x . | \(f(x)=2 a^2+3 a-1\) |
| Simplify. |
For the function \(f(x)=3x^2−2x+1\), find the following:
- \(f(3)\)
- \(f(−1)\)
- \(f(t)\)
- Answer
-
a. \(f(3)=22\)
b. \(f(−1)=6\)
c. \(f(t)=3t^2−2t+1\)
For the function \(g(x)=3x−5\), evaluate the function.
- \(g(h^2)\)
- \(g(x+2)\)
- \(g(x)+g(2)\)
a.
| \(g(x)=3 x-5\) | |
| To evaluate \(g(h^2)\), substitute \(h^2\) for x . | \(g\left(h^2\right)=3 h^2-5\) |
| \(g\left(h^2\right)=3 h^2-5\) |
b.
| \(g(x)=3 x-5\) | |
| To evaluate \(g(x+2)\), substitute \(x+2\) for x . | \(g(x+2)=3(x+2)-5\) |
| Simplify. | \(g(x+2)=3 x+6-5\) |
| \(g(x+2)=3 x+1\) |
c.
| \(g(x)=3 x-5\) | |
| To evaluate \(g(x)+g(2)\), first find \(g(2)\). | \(g(2)=3 \cdot 2-5\) |
| \(g(2)=1\) | |
| \(g(x)+g(2)=\underbrace{3 x-5}_{g(x) \quad \quad g(2)}+1\) | |
| Simplify. | \(g(x)+g(2)=3 x-5+1\) |
| \(g(x)+g(2)=3 x-4\) |
Notice the difference between part b and c. We get \(g(x+2)=3x+1\) and \(g(x)+g(2)=3x−4\). So we see that \(g(x+2)\neq g(x)+g(2)\).
For the function \(g(x)=4x−7\), evaluate the function.
- \(g(m^2)\)
- \(g(x−3)\)
- \(g(x)−g(3)\)
- Answer
-
a. \(4m^2−7\)
b. \(4x−19\)
c. \(4x−12\)
Given the function \(g(m)=\sqrt{m−4}\), evaluate
- \(g(5)\)
- \(g(7)\)
- Answer
-
- \(g(5)=1\)
- \(g(7)=\sqrt{3}\)
Note: \(\sqrt{3}\) is an irrational number. Since the directions did not indicate to approximate, we leave the answer in this form, called exact form , since irrational numbers are non-terminating non-repeating decimals. If the directions indicated to round to one decimal place, then we could state the answer as an approximatimation as, \(g(7)\approx1.7\). Do remember to round appropriately.
Evaluate \(f(x)=x^2+3x−4\) at
- \(2\)
- \(a\)
- \(a+h\)
- Evaluate \(\frac{f(a+h)−f(a)}{h}\)
Solution
Replace the x in the function with each specified value.
a. Because the input value is a number, 2, we can use simple algebra to simplify.
\[\begin{align*}f(2)&=2^2+3(2)−4\\&=4+6−4\\ &=6\end{align*}\]
b. In this case, the input value is a letter so we cannot simplify the answer any further.
\[f(a)=a^2+3a−4\nonumber\]
c. With an input value of \(a+h\), we must use the distributive property.
\[\begin{align*}f(a+h)&=(a+h)^2+3(a+h)−4\\&=a^2+2ah+h^2+3a+3h−4 \end{align*}\]
d. In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that
\[f(a+h)=a^2+2ah+h^2+3a+3h−4\nonumber\]
and we know that
\[f(a)=a^2+3a−4 \nonumber\]
Now we combine the results and simplify.
\[\begin{align*}\dfrac{f(a+h)−f(a)}{h}&=\dfrac{(a^2+2ah+h^2+3a+3h−4)−(a^2+3a−4)}{h}\\ &=\dfrac{(2ah+h^2+3h)}{h} \\ &=\dfrac{h(2a+h+3)}{h} & &\text{Factor out h.}\\ &=2a+h+3 & & \text{Simplify.}\end{align*}\]
In the previous example, the formula in part d is called a difference quotient . In calculus, the difference quotient provides us with a way to describe instantaneous rate of change which is very useful.
Many everyday situations can also be modeled using functions. Let's look at an example.
The number of unread emails in Sylvia’s account is 75. This number grows by 10 unread emails a day. The function \(N(t)=75+10t\) represents the relation between the number of emails, N , and the time, t , measured in days.
- Determine the independent and dependent variable.
- Find \(N(5)\). Explain what this result means.
- Answer
-
a. The number of unread emails is a function of the number of days. The number of unread emails, N , depends on the number of days, t . Therefore, the variable N , is the dependent variable and the variable tt is the independent variable.
b. Find \(N(5)\). Explain what this result means.
\(N(t)=75+10 t\) Substitute in t=5.t=5. \(N({\color{Red} 5})=75+10 \cdot {\color{Red} 5}\) Simplify. \(N(5)=75+50\) \(N(5)=125\) Since 5 is the number of days, \(N(5)\), is the number of unread emails after 5 days. After 5 days, there are 125 unread emails in the account.
Finding Input Value(s) When Given an Output Value
When we know an output value and want to determine the input values that would produce the given output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.
Given the function \(h(p)=p^2+2p\), solve for \(h(p)=3\).
Solution
\[\begin{array}{rl} h(p)=3\\p^2+2p=3 & \text{Substitute the original function}\\ p^2+2p−3=0 & \text{Subtract 3 from each side.}\\(p+3)(p−1)=0&\text{Factor.}\end{array} \nonumber \]
If \((p+3)(p−1)=0\), either \((p+3)=0\) or \((p−1)=0\) (or both of them equal \(0\)). We will set each factor equal to \(0\) and solve for \(p\) in each case.
\[(p+3)=0,\; p=−3 \nonumber \]
\[(p−1)=0,\, p=1 \nonumber\]
This gives us two solutions. The output \(h(p)=3\) when the input is either \(p=1\) or \(p=−3\). We can also verify by graphing as in Figure \(\PageIndex{6}\). The graph verifies that \(h(1)=h(−3)=3\) and \(h(4)=24\).
Given the function \(g(m)=\sqrt{m−4}\), solve \(g(m)=2\).
- Answer
-
\(m=8\)
Writing Equations as Functions
Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can write the equation using function notation. For example, the equation \(2n+6p=12\) represents a functional relationship between \(n\) and \(p\). We can rewrite the equation as \(p\) is a function of \(n\) by solving the equation for \(p\).
Express the relationship \(2n+6p=12\) as a function \(p=f(n)\)
Solution
To express the relationship in this form, we need to write the relationship where \(p\) is a function of \(n\), which means writing it as \(p=[\text{expression involving }n]\).
\[\begin{align*}2n+6p&=12 \\ 6p&=12−2n && \text{Subtract 2n from both sides.} \\ p&=\dfrac{12−2n}{6} & &\text{Divide both sides by 6 and simplify.} \\ p&=\frac{12}{6}−\frac{2n}{6} \\ p&=2−\frac{1}{3}n\end{align*}\]
Therefore, \(p\) as a function of \(n\) is written as
\[p=f(n)=2−\frac{1}{3}n \nonumber\]
Note: Not every relationship can be expressed using function notation. This can only be done when the equation represents a function.
Piecewise Defined Functions
Piecewise defined functions are used in many real-world phenomena (e.g. postal rates and income tax formulas ) are modeled by such functions. It is important that we are familiar with them and know how to evaluate them.
Consider the absolute value function \(f(x)=\left|x\right|\).
The domain is real numbers and the range is values greater than or equal to 0.
If we input 0 or any positive value, the output is the same value as the input (\(f(0)=\left|0\right|=0\), \(f(3)=\left|3\right|=3\), \(f(7.2)=\left|7.2\right|=7.2\), ...).
\(f(x) = x\) if \(x \ge 0\)
If we input a negative value the sign of the output must change from negative to positive (\(f(-1)=\left|-1\right|=1\), \(f(-4.5)=\left|-4.5\right|=4.5\), ...).
\(f(x) = -x\) if \(x < 0\), since multiplying a negative value by -1 makes it positive.
Since this requires two different processes or pieces, the absolute value function is often called the most basic piecewise defined function.
A piecewise function is a function in which the formula used depends upon the domain the input lies in. We notate this idea like:
\[f(x) = \begin{cases} \text{formula 1, if domain value satisfies given criteria 1} \\ \text{formula 2, if domain value satisfies given criteria 2} \\ \text{formula 3, if domain value satisfies given criteria 3} \end{cases}\nonumber \]
Note that piecewise defined functions are not rescricted to 3 formulas, they may contain 2 or more.
A museum charges $5 per person for a guided tour with a group of 1 to 9 people, or a fixed $50 fee for 10 or more people in the group. Set up a function relating the number of people, \(n\), to the cost, \(C\).
Solution
To set up this function, two different formulas would be needed. C = 5 n would work for n values under 10, and C = 50 would work for values of n ten or greater. Notating this:
\[C(n) = \begin{cases} 5n & \text{if 0 < n < 10} \\ 50 &\text{if } n \ge 10\end{cases}\nonumber \]
For the function \(f(x)=\begin{cases}x+6, & \text{if }x<1\\ 3-x^2, &\text{if }x≥1\end{cases}\), find the following:
- \(f(0)\)
- \(f(1)\)
- \(f(3)\)
Solution
- Since the input is 0, we input into the top formula, \(x+6\). So, \(f(0)=0+6=6\)
- Since the input is 1, we input into the top formula, \(x+6\). So, \(f(1)=1+6=7\)
- Since the input is 3 we input into the bottom formula, \(3-x^2\). So, \(f(0)=0+6=6\)
A cell phone company uses the function below to determine the cost, C , in dollars for g gigabytes of data transfer.
\[C(g)=\left\{\begin{array}{ccc} {25} & \text{if} & {0<g<2} \\ {25+10(g-2)} & \text{if} & {g\ge 2} \end{array}\right.\nonumber \]
Find the cost of using 1.5 gigabytes of data, and the cost of using 4 gigabytes of data.
Solution
To find the cost of using 1.5 gigabytes of data, \(C(1.5)\), we first look to see which piece of domain our input falls in. Since 1.5 is less than 2, we use the first formula, giving \(C(1.5)\) = $25.
The find the cost of using 4 gigabytes of data, \(C(4)\), we see that our input of 4 is greater than 2, so we’ll use the second formula. \(C(4)\) = 25 + 10(4 - 2) = $45.
The height \(h\) in feet of a model rocket above the ground \(t\) seconds after lift-off is given by
\[ h(t) = \left\{ \begin{array}{rcl} -5t^2 + 100t, & \mbox{if} & 0 \leq t \leq 20 \\ 0, & \mbox{if} & t > 20 \\ \end{array} \right.\nonumber\]
- Find and interpret \(h(10)\) and \(h(60)\).
- Solve \(h(t) = 375\) and interpret your answers.
Solution
- We first note that the independent variable here is \(t\), chosen because it represents time. Secondly, the function is broken up into two rules: one formula for values of \(t\) between \(0\) and \(20\) inclusive, and another for values of \(t\) greater than 20. Since \(t=10\) satisfies the inequality \(0 \leq t \leq 20\), we use the first formula listed, \(h(t) = -5t^2 + 100t\), to find \(h(10)\). We get \(h(10) = -5(10)^2 + 100(10) = 500\). Since \(t\) represents the number of seconds since lift-off and \(h(t)\) is the height above the ground in feet, the equation \(h(10) = 500\) means that \(10\) seconds after lift-off, the model rocket is \(500\) feet above the ground. To find \(h(60)\), we note that \(t=60\) satisfies \(t > 20\), so we use the rule \(h(t) = 0\). This function returns a value of \(0\) regardless of what value is substituted in for \(t\), so \(h(60) = 0\). This means that \(60\) seconds after lift-off, the rocket is \(0\) feet above the ground; in other words, a minute after lift-off, the rocket has already returned to Earth.
- Since the function \(h\) is defined in pieces, we need to solve \(h(t) = 375\) in pieces. For \(0 \leq t \leq 20\), \(h(t) = -5t^2 + 100t\), so for these values of \(t\), we solve \(-5t^2 + 100t = 375\). Rearranging terms, we get \(5t^2 - 100t + 375 = 0\), and factoring gives \(5(t-5)(t-15) = 0\). Our answers are \(t=5\) and \(t=15\), and since both of these values of \(t\) lie between \(0\) and \(20\), we keep both solutions. For \(t>20\), \(h(t) = 0\), and in this case, there are no solutions to \(0=375\). In terms of the model rocket, solving \(h(t) = 375\) corresponds to finding when, if ever, the rocket reaches \(375\) feet above the ground. Our two answers, \(t=5\) and \(t=15\) correspond to the rocket reaching this altitude twice - once \(5\) seconds after launch, and again \(15\) seconds after launch.
Domain of Functions
We have learned that the domain of a function is the set of input values. How do find the domain when we are given a formula for a function?
First, let's recall some things from algebra.
If \(f(x) = \frac{x}{x - 5}\), find \(f(5)\). Evaluating this,
\[f(5) = \dfrac{(5)}{(5) - 5} = \dfrac{5}{0},\nonumber \]
which is undefined. Since the number \(5\) makes the function undefined, this value is excluded from the domain. Since \(5\) is the only real number that would make the denominator zero, the domain is \((-\infty, 5) \cup (5, \infty)\).
Next, If \(g(x) = \sqrt{x + 3}\), find \(g(-4)\). Evaluating this,
\[g(4) = \sqrt{(-4) + 3} = \sqrt{-1}\nonumber \]
which is not a real number.
Any real number that we input into this function that makes \(x + 3\) negative must be excluded from the domain. So, the domain of this function is all real numbers such that \(x + 3 \geq 0\) which yields \(x \geq -3\). The domain in interval notation is \([-3, \infty)\).
When finding the domain of a function, we start with the set of all real numbers. Then, we exclude any values that make a denominator zero or result in an even root of a negative number.
- Identify the input values.
- Identify any restrictions on the input and exclude those values from the domain. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for x . If the function’s formula contains an even root, set the radicand greater than or equal to 0, and then solve.
- Write the domain in interval notation.
Find the domain of the function \(f(x)=x^2−1\).
Solution
The input value, shown by the variable x in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.
In interval form, the domain of f is \((−\infty,\infty)\).
Find the domain of the function:
\[f(x)=5−x+x^3 \nonumber\]
- Answer
-
\((−\infty,\infty)\)
Find the domain of the function \(f(x)=\dfrac{x+1}{2−x}\).
Solution
When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for x.
\[ \begin{align*} 2−x=0 \\[4pt] −x &=−2 \\[4pt] x&=2 \end{align*}\]
Now, we will exclude 2 from the domain. The answers are all real numbers where \(x<2\) or \(x>2\). We can use a symbol known as the union, \(\cup\),to combine the two sets. In interval notation, we write the solution:\((−\infty,2)∪(2,\infty)\).
In interval notation, the domain of f is \((−\infty,2)\cup(2,\infty)\).
Find the domain of the function:
\[f(x)=\dfrac{1+4x}{2x−1} \nonumber\]
- Answer
-
\[(−\infty,\dfrac{1}{2})\cup(\dfrac{1}{2},\infty) \nonumber\]
Find the domain of the function:
\[f(x)=\sqrt{7-x} \nonumber \]
Solution
When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.
Set the radicand greater than or equal to zero and solve for x.
\[ \begin{align*} 7−x&≥0 \\[4pt] −x&≥−7\\[4pt] x&≤7 \end{align*}\]
Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to 7, or \(\left(−\infty,7\right]\).
Find the domain of the function
\[f(x)=\sqrt{5+2x}. \nonumber\]
- Answer
-
\[\left[−2.5,\infty\right) \nonumber\]
Find the domain of the following functions. State answer in interval notation.
- \(g(x) = \sqrt{4 - 3x}\)
- \(h(x) = \sqrt[5]{4 - 3x}\)
- \(R(x) = \sqrt{x^2+5}\)
- \( F(x) = \dfrac{\sqrt[4]{2x+1}}{x^2 - 1}\)
- \(r(t) = \dfrac{4}{6 - \sqrt{t+3}}\)
- \( I(p) = \dfrac{3p^2}{p}\)
Solution
- Set \(4 - 3x \geq 0\). Solving this, we get \(x \leq \frac{4}{3}\). This shows that as long as \(x \leq \frac{4}{3}\), the radicand is greater than or equal to \(0\), \(4 - 3x \geq 0\), and the formula \(g(x)\) outputs a real number. The domain is \(\left(-\infty, \frac{4}{3}\right]\).
- Although the formula for \(h(x)\) is very similar to that of \(g(x)\) there is one key difference: \(g(x)\) is an even indexed root (namely a square root) and \(h(x)\) is an odd indexed root (the fifth root). Since odd roots of real numbers (even negative real numbers) are real numbers, there is no restriction on the inputs to \(h\). Hence, the domain is \((-\infty, \infty)\).
- Since \(x^2+5>0\) for all x, there is no restrictions on the inputs. Hence, the domain is \((-\infty, \infty)\).
- There are two issues here: not only do we have a denominator, we have a fourth (even-indexed) root. We must determine the restrictions on each part and select the real numbers which satisfy both conditions. To satisfy the fourth root, \(2x+1 \geq 0\). From this we get \(x \geq -\frac{1}{2}\). Next, we find the values of \(x\) which make the denominator equal to \(0\), \(x^2 - 1=0\). Solving this, \(x = \pm 1\). Hence, in order for a real number \(x\) to be in the domain of \(F\), \(x \geq -\frac{1}{2}\) but \(x \neq \pm 1\). In interval notation, this set is \(\left[ -\frac{1}{2}, 1 \right) \cup (1, \infty)\).
- Don't be put off by the '\(t\)' here. It is an independent variable representing a real number, just like \(x\) does, and is subject to the same restrictions. As in the previous problem, we have two issues here: we have a square root and a denominator. To satisfy the square root, we need a non-negative radicand so we set \(t + 3 \geq 0\) to get \(t \geq -3\). Setting the denominator equal to zero gives \(6 - \sqrt{t+3} =0\), or \(\sqrt{t+3} = 6\). Squaring both sides gives \(t+3 = 36\), or \(t = 33\). Remember that since we squared both sides in the course of solving this equation, we need to check our answer. Sure enough, when \(t=33\), \(6 - \sqrt{t+3} = 6 - \sqrt{36} = 0\), so \(t=33\) will cause problems in the denominator. For the domain of \(r\): we need \(t \geq -3\), but \(t \neq 33\). Hence, the domain is \([-3, 33) \cup (33, \infty)\).
- It's tempting to simplify \(I(x) = \frac{3x^2}{x} = 3x\). Since there are no longer any denominators it appears that the domain is \((-\infty, \infty)\) which is not the case. When simplifying \(I(x)\), we are assuming \(x \neq 0\), since \(\frac{0}{0}\) is undefined. Proceeding as before, we find the domain of \(I\) to be all real numbers except \(0\): \((-\infty, 0) \cup (0, \infty)\).
It is worth reiterating the importance of finding the domain of a function before simplifying, as evidenced by the function \(I\) in the previous example. Even though the formula \(I(x)\) simplifies to \(3x\), it would be inaccurate to write \(I(x) = 3x\) without adding the stipulation that \(x \neq 0\).
Find the domain of the function \(v(t)=\sqrt{6-5t-t^{2} }\).
Solution
A square root is only defined when the quantity inside the square root, is zero or greater. Thus, the domain of this function will be when \(6-5t-t^{2} \ge 0\). Solving this non-linear inequality yields \([-6, 1]\) which is the domain of the \(v(t)\) function.
Given \(g(x)=\dfrac{4+x}{x-3}\),
- Find and simplify \(g\left(\dfrac{1}{t}\right)\).
- What is the domain of the function you found in part 1? State your answer in interval notation.
Solution
- \[\begin{align*}g\left(\dfrac{1}{t}\right)&=\dfrac{4+\dfrac{1}{t}}{\dfrac{1}{t}-3}\\[4pt] &=\left(\dfrac{t}{t}\right)\left(\dfrac{4+\dfrac{1}{t}}{\dfrac{1}{t}-3}\right)\\[4pt] &=\dfrac{4t+1}{1-3t} \end{align*}\]
- For \(g\left(\dfrac{1}{t}\right) \text{ to be defined, } t\neq0\). In addition, for the function to be defined, \(1-3t\neq0\). So, \(t\neq\dfrac{1}{3}\).
The domain is \((-\infty,0) \cup \left(0,\dfrac{1}{3}\right) \cup \left(\dfrac{1}{3},\infty\right)\).
Given \(f(a)=\dfrac{3a}{a^2-7a+10} \text{ and } g(a)=\dfrac{2a}{a^2-8a+15}\),
- Find and simplify \(f(a)-g(a)\).
- What is the domain of the function you found in part 1? State your answer in interval notation.
- Answer
-
- \(\dfrac{a}{(a-3)(a-2)}\)
- \((-\infty,2) \cup (2,3) \cup (3,5) \cup (5,\infty)\)


Despite its unsaturation, the benzene ring is inert to strong oxidizing agents such as KMnO
4
, which will cleave alkene carbon–carbon bonds (Section 8.8). It turns out, however, that the presence of the aromatic ring has a dramatic effect on the reactivity of alkyl side chains. These side chains react rapidly with oxidizing agents and are converted into carboxyl groups, –CO
2
H.The net effect is conversion of an alkylbenzene into a benzoic acid,
2.8: Reactions at the Benzylic Position
After completing this section, you should be able to
- write an equation to describe the oxidation of an alkylbenzene to a carboxylic acid.
- identify the reagents required to oxidize a given alkylbenzene to a carboxylic acid.
- identify the product formed from the side-chain oxidation of a given alkylbenzene.
- identify the aromatic compound needed to produce a given carboxylic acid through side-chain oxidation.
- write the equation for the bromination of an alkylbenzene side chain.
- identify the reagents and conditions necessary to bring about bromination in the side chain of an alkylbenzene.
- identify the product formed when a given alkylbenzene undergoes side-chain bromination.
- identify the alkylbenzene needed to prepare a given benzylic bromide by radical substitution.
- write the mechanism for the radical substitution at the benzylic position of an alkylbenzene.
- explain the stability of benzylic radicals in terms of resonance, and draw the resonance contributors of a given benzyl radical.
- explain, and illustrate with appropriate examples, the importance of benzylic bromides as intermediates in organic syntheses.
- arrange a given series of radicals (including benzylic type radicals) in order of increasing or decreasing stability. (Review Section 10.3 if necessary.)
Make certain that you can define, and use in context, the key terms below.
- benzylic oxidation
- benzylic position
- side-chain oxidation
As you can see from the examples, no matter what the length of the alkyl group in the arene substrate, the product is always a one-carbon carboxyl group. Thus, the benzylic carbon atom has been oxidized and the term benzylic oxidation is appropriate. The term side-chain oxidation is also commonly used.
In alkylbenzenes, the carbon atom which is attached to the aromatic ring is particularly reactive. Reactions taking place at this carbon atom are said to occur at the benzylic position .
You may wish to review allylic bromination using N-bromosuccinimide from CHEM 231.
Benzylic halides undergo the typical reactions of alkyl halides, so they are frequently used in multistep syntheses.
Note that we have adopted the terminology given below.
|
Any compound of the type
(where X = halogen) will be referred to as a “benzylic halide.” |
Compounds of the type
are actually called benzyl chloride, benzyl bromide, etc. |
The compound
is called benzyl alcohol. |
Oxidation of Alkyl Side-Chains
Two other examples of this reaction are given below, and illustrate its usefulness in preparing substituted benzoic acids.
The mechanism of side-chain oxidation is complex and involves reaction of C–H bonds at the position next to the aromatic ring to form intermediate benzylic radicals. tert -Butylbenzene has no benzylic hydrogens, however, and is therefore inert.
Predict the products of the following two reactions.
- Answer
-
The second one leads to no reaction because it requires a hydrogen just off the phenyl ring.
What aromatic products would you obtain from the KMnO 4 oxidation of the following substances?
a. b.
- Answer
-
- m -Nitrobenzoic acid
- p - tert -Butylbenzoic acid
Bromination of the Benzylic Carbon
Side-chain bromination at the benzylic position occurs when an alkylbenzene is treated with N -bromosuccinimide (NBS). For example, propylbenzene gives (1-bromopropyl)benzene in 97% yield on reaction with NBS in the presence of benzoyl peroxide, (PhCO 2 ) 2 , as a radical initiator. Bromination occurs exclusively in the benzylic position next to the aromatic ring and does not give a mixture of products.
The mechanism of benzylic bromination is similar to that discussed in Section 10.3 for allylic bromination of alkenes. Abstraction of a benzylic hydrogen atom first generates an intermediate benzylic radical, which then reacts with Br 2 in step 2 to yield product and a radical, which cycles back into the reaction to carry on the chain shown below as a summary. The Br 2 needed for reaction with the benzylic radical is produced in step 3 by a concurrent reaction of HBr with NBS.
Reaction occurs exclusively at the benzylic position because the benzylic radical intermediate is stabilized by resonance. Figure \(\PageIndex{1}\) shows how the benzyl radical is stabilized by overlap of its p orbital with the ringed \(\pi\) electron system.
Consider a benzyl radical. Would it be more stable than an alkyl radical? Explain.
- Answer
-
Yes, it would be more stable than an alkyl radical. The benzyl radical is stabilized through several resonance structures where the radical is moved through the ring via the pi system there.
How would you make the following molecule?
- Answer
-
The following is just one possibility.


4: Diffraction
In the preceding chapter, we implicitly regarded slits as objects with positions but no size. The widths of the slits were considered negligible. When the slits have finite widths, each point along the opening can be considered a point source of light—a foundation of Huygens’s principle. Because real-world optical instruments must have finite apertures (otherwise, no light can enter), diffraction plays a major role in the way we interpret the output of these optical instruments. For example, diffraction places limits on our ability to resolve images or objects. This is a problem that we will study later in this chapter.
-
- 4.1: Prelude to Diffraction
- Due to Huygens’s principle, we can imagine a wave front as equivalent to inﬁnitely many point sources of waves. Thus, a wave from a slit can behave not as one wave but as an infinite number of point sources. These waves can interfere with each other, resulting in an interference pattern without the presence of a second slit. This phenomenon is called diffraction.
-
- 4.3: Intensity in Single-Slit Diffraction
- The intensity pattern for diffraction due to a single slit can be calculated using phasors as \(I = I_0 \left(\frac{sin \space \beta}{\beta}\right)^2,\) where \(\beta = \frac{\phi}{2} = \frac{\pi D \space sin \space \theta}{\lambda}\), D is the slit width, λλ is the wavelength, and θθ is the angle from the central peak.
-
- 4.4: Double-Slit Diffraction
- With real slits with finite widths, the effects of interference and diffraction operate simultaneously to form a complicated intensity pattern. Relative intensities of interference fringes within a diffraction pattern can be determined. Missing orders occur when an interference maximum and a diffraction minimum are located together.
-
- 4.5: Diffraction Gratings
- A diffraction grating consists of a large number of evenly spaced parallel slits that produce an interference pattern similar to but sharper than that of a double slit. Constructive interference occurs when \(d \space sin \space \theta = m \lambda\) form = 0, ± 1, ±2,..., where d is the distance between the slits, θ is the angle relative to the incident direction, and m is the order of the interference.
-
- 4.6: Circular Apertures and Resolution
- Light diffracts as it moves through space, bending around obstacles, interfering constructively and destructively. This can be used as a spectroscopic tool—a diffraction grating disperses light according to wavelength, for example, and is used to produce spectra—but diffraction also limits the detail we can obtain in images.Diffraction limits the resolution in many situations. The acuity of our vision is limited because light passes through the pupil, which is the circular aperture of the eye.
-
- 4.7: X-Ray Diffraction
- Since X-ray photons are very energetic, they have relatively short wavelengths. Thus, typical X-ray photons act like rays when they encounter macroscopic objects, like teeth, and produce sharp shadows. However, since atoms are on the order of 0.1 nm in size, X-rays can be used to detect the location, shape, and size of atoms and molecules. The process is called X-ray diffraction, and it involves the interference of X-rays to produce patterns.
-
- 4.8: Holography
- A hologram is a true three-dimensional image recorded on film by lasers. Holograms are used for amusement; decoration on novelty items and magazine covers; security on credit cards and driver’s licenses (a laser and other equipment are needed to reproduce them); and for serious three-dimensional information storage. You can see that a hologram is a true three-dimensional image because objects change relative position in the image when viewed from different angles.
Thumbnail: X-ray diffraction from the crystal of a protein (hen egg lysozyme) produced this interference pattern. Analysis of the pattern yields information about the structure of the protein. (credit: “Del45”/Wikimedia Commons)


6.3: Shortest Path
When you visit a website like Google Maps or use your Smartphone to ask for directions from home to your Aunt’s house in Pasadena, you are usually looking for a shortest path between the two locations. These computer applications use representations of the street maps as graphs, with estimated driving times as edge weights.
While often it is possible to find a shortest path on a small graph by guess-and-check, our goal in this chapter is to develop methods to solve complex problems in a systematic way by following algorithms . An algorithm is a step-by-step procedure for solving a problem. Dijkstra’s (pronounced dike-stra) algorithm will find the shortest path between two vertices.
- Mark the ending vertex with a distance of zero. Designate this vertex as current.
- Find all vertices leading to the current vertex. Calculate their distances to the end. Since we already know the distance the current vertex is from the end, this will just require adding the most recent edge. Don’t record this distance if it is longer than a previously recorded distance.
- Mark the current vertex as visited. We will never look at this vertex again.
- Mark the vertex with the smallest distance as current, and repeat from step 2.
Suppose you need to travel from Tacoma, WA (vertex T) to Yakima, WA (vertex Y). Looking at a map, it looks like driving through Auburn (A) then Mount Rainier (MR) might be shortest, but it’s not totally clear since that road is probably slower than taking the major highway through North Bend (NB). A graph with travel times in minutes is shown below. An alternate route through Eatonville (E) and Packwood (P) is also shown.
Solution
Step 1: Mark the ending vertex with a distance of zero. The distances will be recorded in [brackets] after the vertex name.
Step 2: For each vertex leading to Y, we calculate the distance to the end. For example, NB is a distance of 104 from the end, and MR is 96 from the end. Remember that distances in this case refer to the travel time in minutes.
Step 3 & 4: We mark Y as visited, and mark the vertex with the smallest recorded distance as current. At this point, P will be designated current. Back to step 2.
Step 2 (#2): For each vertex leading to P (and not leading to a visited vertex) we find the distance from the end. Since E is 96 minutes from P, and we’ve already calculated P is 76 minutes from Y, we can compute that E is 96+76 = 172 minutes from Y.
If we make the same computation for MR, we’d calculate 76+27 = 103. Since this is larger than the previously recorded distance from Y to MR, we will not replace it.
Step 3 & 4 (#2): We mark P as visited, and designate the vertex with the smallest recorded distance as current: MR. Back to step 2.
Step 2 (#3): For each vertex leading to MR (and not leading to a visited vertex) we find the distance to the end. The only vertex to be considered is A, since we’ve already visited Y and P. Adding MR’s distance 96 to the length from A to MR gives the distance 96+79 = 175 minutes from A to Y.
Step 3 & 4 (#3): We mark MR as visited, and designate the vertex with smallest recorded distance as current: NB. Back to step 2.
Step 2 (#4): For each vertex leading to NB, we find the distance to the end. We know the shortest distance from NB to Y is 104 and the distance from A to NB is 36, so the distance from A to Y through NB is 104+36 = 140. Since this distance is shorter than the previously calculated distance from Y to A through MR, we replace it.
Step 3 & 4 (#4): We mark NB as visited, and designate A as current, since it now has the shortest distance.
Step 2 (#5): T is the only non-visited vertex leading to A, so we calculate the distance from T to Y through A: 20+140 = 160 minutes.
Step 3 & 4 (#5): We mark A as visited, and designate E as current.
Step 2 (#6): The only non-visited vertex leading to E is T. Calculating the distance from T to Y through E, we compute \(172+57 = 229\) minutes. Since this is longer than the existing marked time, we do not replace it.
Step 3 (#6): We mark E as visited. Since all vertices have been visited, we are done.
From this, we know that the shortest path from Tacoma to Yakima will take 160 minutes. Tracking which sequence of edges yielded 160 minutes, we see the shortest path is T-A-NB-Y.
Dijkstra’s algorithm is an optimal algorithm , meaning that it always produces the actual shortest path, not just a path that is pretty short, provided one exists. This algorithm is also efficient , meaning that it can be implemented in a reasonable amount of time. Dijkstra’s algorithm takes around V 2 calculations, where V is the number of vertices in a graph[1]. A graph with 100 vertices would take around 10,000 calculations. While that would be a lot to do by hand, it is not a lot for computer to handle. It is because of this efficiency that your car’s GPS unit can compute driving directions in only a few seconds.
In contrast, an inefficient algorithm might try to list all possible paths then compute the length of each path. Trying to list all possible paths could easily take 10 25 calculations to compute the shortest path with only 25 vertices; that’s a 1 with 25 zeros after it! To put that in perspective, the fastest computer in the world would still spend over 1000 years analyzing all those paths.
A shipping company needs to route a package from Washington, D.C. to San Diego, CA. To minimize costs, the package will first be sent to their processing center in Baltimore, MD then sent as part of mass shipments between their various processing centers, ending up in their processing center in Bakersfield, CA. From there it will be delivered in a small truck to San Diego.
The travel times, in hours, between their processing centers are shown in the table below. Three hours has been added to each travel time for processing. Find the shortest path from Baltimore to Bakersfield.
Solution
While we could draw a graph, we can also work directly from the table.
Step 1: The ending vertex, Bakersfield, is marked as current.
Step 2: All cities connected to Bakersfield, in this case Denver and Dallas, have their distances calculated; we’ll mark those distances in the column headers.
Step 3 & 4: Mark Bakersfield as visited. Here, we are doing it by shading the corresponding row and column of the table. We mark Denver as current, shown in bold, since it is the vertex with the shortest distance.
Step 2 (#2): For cities connected to Denver, calculate distance to the end. For example, Chicago is 18 hours from Denver, and Denver is 19 hours from the end, the distance for Chicago to the end is 18+19 = 37 (Chicago to Denver to Bakersfield). Atlanta is 24 hours from Denver, so the distance to the end is 24+19 = 43 (Atlanta to Denver to Bakersfield).
Step 3 & 4 (#2): We mark Denver as visited and mark Dallas as current.
Step 2 (#3): For cities connected to Dallas, calculate the distance to the end. For Chicago, the distance from Chicago to Dallas is 18 and from Dallas to the end is 25, so the distance from Chicago to the end through Dallas would be 18+25 = 43. Since this is longer than the currently marked distance for Chicago, we do not replace it. For Atlanta, we calculate 15+25 = 40. Since this is shorter than the currently marked distance for Atlanta, we replace the existing distance.
Step 3 & 4 (#3): We mark Dallas as visited, and mark Chicago as current.
Step 2 (#4): Baltimore and Atlanta are the only non-visited cities connected to Chicago. For Baltimore, we calculate 15+37 = 52 and mark that distance. For Atlanta, we calculate 14+37 = 51. Since this is longer than the existing distance of 40 for Atlanta, we do not replace that distance.
Step 3 & 4 (#4): Mark Chicago as visited and Atlanta as current.
Step 2 (#5): The distance from Atlanta to Baltimore is 14. Adding that to the distance already calculated for Atlanta gives a total distance of 14+40 = 54 hours from Baltimore to Bakersfield through Atlanta. Since this is larger than the currently calculated distance, we do not replace the distance for Baltimore.
Step 3 & 4 (#5): We mark Atlanta as visited. All cities have been visited and we are done.
The shortest route from Baltimore to Bakersfield will take 52 hours, and will route through Chicago and Denver.
Find the shortest path between vertices A and G in the graph below.
- Answer
-
The shortest path is ABDEG, with length 13
[1] It can be made to run faster through various optimizations to the implementation.


16.2: Integer Data Type
-
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Overview
An integer data type represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are commonly represented in a computer as a group of binary digits (bits). The size of the grouping varies so the set of integer sizes available varies between different types of computers and different programming languages. [1]
Discussion
The integer data type basically represents whole numbers (no fractional parts). The integer values jump from one value to another. There is nothing between 6 and 7. It could be asked why not make all your numbers floating point which allow for fractional parts. The reason is threefold. First, some things in the real world are not fractional. A dog, even with only 3 legs, is still one (1) dog not ¾ of a dog. Second, the integer data type is often used to control program flow by counting, thus the need for a data type that jumps from one value to another. Third, integer processing is significantly faster within the CPU than is floating point processing.
The integer data type has similar attributes and acts or behaves similarly in all programming languages that support it.
| Language | Reserved Word | Size | Range |
|---|---|---|---|
| C++ |
short
|
16 bits / 2 bytes | -32,768 to32,767 |
| C++ |
int
|
varies | depends on compiler |
| C++ |
long
|
32 bits / 4 bytes | -2,147,483,648 to 2, 147,483,647 |
| C++ |
long long
|
64 bits / 8 bytes | −9,223,372,036,854,775,808 to 9,223,372,036,854,775,807 |
| C# |
short
|
16 bits / 2 bytes | -32,768 to32,767 |
| C# |
int
|
32 bits / 4 bytes | -2,147,483,648 to 2, 147,483,647 |
| C# |
long
|
64 bits / 8 bytes | −9,223,372,036,854,775,808 to 9,223,372,036,854,775,807 |
| Java |
short
|
16 bits / 2 bytes | -32,768 to32,767 |
| Java |
int
|
32 bits / 4 bytes | -2,147,483,648 to 2, 147,483,647 |
| Java |
long
|
64 bits / 8 bytes | −9,223,372,036,854,775,808 to 9,223,372,036,854,775,807 |
| JavaScript | N/A | ||
| Python |
int()
|
no limit | |
| Swift |
Int
|
varies | depends on platform |
| Swift |
Int32
|
32 bits / 4 bytes | -2,147,483,648 to 2, 147,483,647 |
| Swift |
Int64
|
64 bits / 8 bytes | −9,223,372,036,854,775,808 to 9,223,372,036,854,775,807 |
For C++ and Swift the size of a default integer varies with the compiler being used and the computer. This effect is known as being machine dependent . These variations of the integer data type are an annoyance for a beginning programmer. For a beginning programmer, it is more important to understand the general attributes of the integer data type that apply to most programming languages.
JavaScript does not support an integer data type, but the
Math.round()
function may be used to return the value of a number rounded to the nearest integer.
[2]
Python 3 integers are not limited in size, however,
sys.maxsize
may be used to determine the maximum practical size of a list or string index.
[3]
Key Terms
- machine dependent
- An attribute of a programming language that changes depending on the computer’s CPU.


16.5: Writing Process- Thinking Critically About Text
By the end of this section, you will be able to:
- Develop a writing project focused on textual analysis.
- Complete the stages of the writing process, including generating ideas, drafting, reviewing, revising, rewriting, and editing.
- Integrate the writer’s ideas with ideas of others.
- Collaborate in the peer review process.
When analyzing a text, writers usually focus on the content of the text itself and deliberately leave themselves in the background, minimizing personal presence and bias. To write this way, they avoid first-person pronouns and value judgments. In reality, of course, writers do reveal their presence by the choices they make: what they include, what they exclude, and what they emphasize. Although your own subjectivity and situation will likely affect your inferences and judgments, recognizing these potential biases will help you keep the focus on your subject and off yourself.
Summary of Assignment
Write an analytical essay about a short story or another short text of your choice, either fiction or literary nonfiction. If desired, you may choose “The Storm” by Kate Chopin, reprinted above. Consider the author’s form and organization, tone, or stylistic choices, including diction and sensory or figurative language. You might also consider the historical or social context, the theme, the character development, or the relation between setting and plot or characterization. If you are free to choose your own text and topic, consider the following approaches:
- Analyze the literary components mentioned and focus your essay on their significance in the work.
- Like student author Gwyn Garrison, choose one or several components and examine how different authors use them and how they relate to broader contexts.
Convincing textual analysis essays usually include the following information:
- overview of the text, identifying author, title, and genre
- very brief summary
- description of the text’s form and structure
- explanation of the author’s point of view
- summary of the social, historical, or cultural context in which the work was written
- assertion or thesis about what the text means: your main task as an analyst
When writing about a novel or short story, explain how the main elements function:
- narrator (who tells the story)
- plot (what happens in the story)
- one or more characters (who are acting or being acted upon)
- setting (when and where things are happening)
- theme (the meaning of the story)
Keep in mind that the author who writes the story is different from the narrator and invented characters in it. Keep in mind, too, that what happens in the story—the plot—is different from the meaning of the story—the theme. Understanding what happens will help you discover what the text means.
The elements ( https://openstax.org/r/elements ) of literary or narrative nonfiction are similar to those of a fictional story except that everything in the text is supposed to have really happened. For this reason, the author and the narrator of the story may be one and the same. Informational nonfiction—essays, reports, and textbook chapters—is also meant to be believed; here, however, ideas and arguments must be strong and well supported to be convincing. When analyzing nonfiction, pay special attention to the author’s thesis or claim and to how it is supported through reasoning and evidence. Also note interesting or unusual tone, style, form, or voice.
Another Lens 1. In writing from a personal or subjective viewpoint, the writer and their beliefs and experiences are necessarily part of your analysis and may need to be expressed and examined. For example, you may write subjectively and compare and contrast your situation with that of the author or a character. You might explain how your personal background causes you to read the text in a particular way that is meaningful to you. If you choose this option, be sure to analyze the text as you would for a more objective analysis before focusing on your personal views.
Another Lens 2. A leading contemporary example of narrative nonfiction writing is Jon Krakauer’s (b. 1954) Into the Wild ( https://openstax.org/r/Into-the-Wild ), the story of Chris McCandless (1968– c. 1992), a young college graduate who lived at subsistence level in the backwoods of Alaska for 113 days. The text is somewhat similar thematically to Henry David Thoreau’s (1817–1862) Walden (1854), written more than a century earlier and discussed later in this section. Both are about dropping out of society to create a meaningful life. After reading the excerpt of Into the Wild linked above, you may choose to write a textual analysis of it either on its own or in light of the sample analysis of Thoreau’s writings later in this section. Consider comparing and contrasting McCandless’s situation with Thoreau’s life in Walden and how Krakauer and Thoreau use various literary elements in their writing. Topics for analysis might be setting, character traits, motives, cultural communities, historical context, and attitudes toward life and society.
Quick Launch: Start with Your Thesis
For textual analysis, your thesis should be a clear, concise statement that identifies your analytical stance on which readers will expect you to elaborate.
Develop a Working Thesis
A working thesis is referred to as such because the thesis is subject to revision. You may have to revisit it later in the writing process, for it is almost impossible to craft a thesis without having analyzed some of the text first. Your thesis, therefore, will come from the element(s) you choose to analyze, such as the following:
- an aspect or several aspects of form and structure and their significance
- social, historical, or cultural context in which the text was written and its significance
- style elements such as diction, imagery, or figurative language and their significance
- aspects of characters, plot, or setting
- overall theme of a single work or more than one work
- comparison or contrast of elements within one or more works
- relation to issues outside the text
To develop a working thesis, use the formula shown in Table \(16.1\), basing your answers on one of the bulleted items listed above.
| What are you doing * Why are you doing it? |
| What are you doing? Analyzing Kate Chopin's short story "The Storm |
| * |
| Why are you doing it? "To argue that..." |
| = |
| In the short story “The Storm,” author Kate Chopin uses linguistic variety to add to the character development. |
You can also start with an analytical question: For what reason(s) does Chopin use linguistic variety? Your initial answer might yield the thesis above. Or you can ask another analytical question, such as this one: In what ways do the plot and setting of “The Storm” reinforce its theme?
Drafting: Explore Possible Areas of Analysis for Fiction: Approach 1
Analytical essays ( https://openstax.org/r/Analytical-essays ) begin by answering basic questions: What genre is this text—poem, play, story, biography, memoir, essay? What is its title? Who is the author? When was it published?
Identify and Summarize the Text
In addition to the basic questions, analytical essays provide a brief summary of the plot or main idea. Summarize briefly, logically, and objectively to provide a background for what you plan to say about the text. This information may be incorporated into the introduction or may follow it.
Explain the Form and Organization
To analyze the organizational structure of a text, ask: How is it put together? Why does the author start here and end there? Why does the author sequence information in this order? What connects the text from start to finish? For example, by repeating words, ideas, and images, writers call attention to these elements and indicate that they are important to the meaning of the text. No matter what the text, some principle or plan holds it together and gives it structure. Fiction and nonfiction texts that tell stories are often, but not always, organized as a sequence of events in chronological order. Poems may have formal structures or other organizational elements. Other texts may alternate between explanations and examples or between firstperson and third-person narrative. You will have to decide which aspects of the text’s form and organization are most important for your analysis.
For example, this student analyzes the point of view of Gwendolyn Brooks’s poem “We Real Cool ( https://openstax.org/r/We-Real-Cool ).”.
Gwendolyn Brooks writes “We Real Cool” (1963) from the point of view of members of a street gang who speak as one voice.   The plural speaker in the poem, “We,” celebrates what adults might call adolescent hedonism—but the speaker, feeling powerful in the group identity, makes a conscious choice for a short, intense life over a long, safe, and dull existence.
Place the Work in Context
To analyze the context of a text, ask: What circumstances (historical, social, political, biographical) produced this text? How does this text compare or contrast with another by the same author or with a similar work by a different author? No text exists in isolation. Each was created by a particular author in a particular place at a particular time. Describing this context provides readers with important background information and indicates which conditions you think were most influential.
For example, this student analyzes the social context of Gwendolyn Brooks’s poem “We Real Cool.”
From society’s viewpoint, the boys are nothing but misfits—refusing to work, leading violent lives, breaking laws, and confronting police. However, these boys live in a society that is dangerous for Black men, who often die at the hands of police even when they are doing the right thing. The boys are hopeless, recognizing no future but death, regardless of their actions, and thus “Die soon.”
Explain the Theme of the Text
To analyze the theme of a text, determine the implied theme in fiction, poetry, and narrative nonfiction. One purpose for writing a textual analysis is to point out the theme. Ask yourself: So what? What is this text really about? What do I think the author is trying to say by writing this text? What problems, puzzles, or ideas are most interesting? In what ways do the characters change between the beginning and end of the text? Good ideas for a thesis arise from material in which the meaning is not obviously stated.
For example, this student analyzes one theme of Gwendolyn Brooks’s poem “We Real Cool.”
For the “Seven at the Golden Shovel,” companionship is everything. For many teenagers, fitting in or conforming to a group identity is more important than developing an individual identity. Brooks expresses this theme through the poem’s point of view, the plural “We” repeated at the end of each line.
Analyze Stylistic Choices
To analyze stylistic choices, examine the details of the text. Ask yourself: Why does the author use this word or phrase instead of a synonym for it? In what ways does this word or phrase relate to other words or phrases? In what ways do the author’s figurative comparisons affect the meaning or tone of the text? In what ways does use of sensory language (imagery) affect the meaning or tone of the text? In what ways does this element represent more than itself? In what ways does the author use sound or rhythm to support meaning?
For example, this student analyzes the diction of Gwendolyn Brooks’s poem “We Real Cool.”
Brooks chooses the word cool to open the poem and build the first rhyme. Being cool is the code by which the boys live. However, the word cool also suggests the idiom “to be placed ‘on ice,’” a term that suggests a delay. The boys live in a state of arrested development, anticipating early deaths. In addition, the term to ice someone means “to kill,” another reference to the death imagery at poem’s end. The boys are not suggesting suicide; they expect to be killed by members of society who find them threatening.
Support Your Analysis
Analytical interpretations ( https://openstax.org/r/Analytical ) are built around evidence from the text itself. You’ll note the quotations in the examples above. Summarize larger ideas in your own language to conserve space. Paraphrase more specific ideas, also in your own words, and quote directly to feature the author’s diction. See Editing Focus: Paragraphs and Transitions and Writing Process: Integrating Research for more information about summarizing, paraphrasing, and quoting directly. If you include outside information for support, comparison, or contrast, document the sources carefully: MLA Documentation and Format.
Use a graphic organizer such as Table \(16.2\) to gather ideas for drafting.
| Genre | Analysis | Your Ideas |
| If you’ve chosen to analyze poetry, |
Consider the role of… Word choice (imagery) Structure of lines and stanzas Sound (meter, rhyme, rhythm) Figurative language (simile, metaphor, personification) |
Why does the author… ? What is the effect on meaning, tone, or audience? |
| If you’ve chosen to analyze drama, |
Consider the role of… Characters and dialogue Setting Plot and conflict Theme Structure of acts, scenes, and stage directions |
Why does the author… ? What is the effect on meaning, tone, or audience? |
| If you’ve chosen to analyze fiction or narrative nonfiction, |
Consider the role of… Characters (narrator or author in narrative nonfiction) Setting Plot (or real events in narrative nonfiction) and conflict Theme Diction Structure (chronology, flashback or forward, foreshadowing, chapters) |
Why does the author… ? What is the effect on meaning, tone, or audience? |
Drafting: Explore Possible Areas of Analysis for Literary Nonfiction: Approach 2
Although similar to fiction, narrative or literary nonfiction has a basic orientation toward exposition: relating real events in a creative way rather than inventing fictional events and characters. In reading and analyzing expository prose, you also may encounter literary language, narrative structure, characters, setting, theme, and plot development, depending on the type of prose. Therefore, your approach to analyzing nonfiction will call on many of the same strategies you use to analyze fiction. Two basic differences, however, are that literary nonfiction may have less dialogue, depending on the genre, and that the author and narrator may be the same. In other words, no intermediary or artistic filter may exist between the author and the work. The nonfiction author is assumed to be speaking a truth, which may be serious, comic, controversial, or neutral. Fictional characters, on the other hand, are creations of an author’s mind; they think and speak as they were created to do.
Planning the Essay
In writing your essay, you will need to present the same kinds of text evidence as you would when analyzing fiction to give credibility to your claims and to support your thesis. And you’ll need to keep in mind the rhetorical situation—purpose, audience, stance, context, and culture—as well, for it remains the building block of an effective analysis. As in most academic essays, body paragraphs refer to the thesis through topic sentences and move consistently toward supporting it before you finally arrive at a convincing conclusion that has grown out of the analysis. In nonfiction, because you assume you are dealing with a truthful explanation of facts and views, your task should be to give a new view and understanding of something that already may be familiar to readers. In writing your analysis, consider the following plan:
- Begin your analysis of nonfiction with an introductory overview in which you include the work’s genre, title, author, and publication date.
- Identify the literary point of view, if relevant: first person— I or plural we —or third-person— he , she , or they .
- Continue with a brief summary of the work, and place it in context: the work’s social, historical, and cultural background will help readers follow your points about its theme.
- Present your thesis near the end of the introduction. It should be argumentative, in an academic sense, so that you can “prove” your points.
- Support your thesis with well-elaborated body paragraphs, as you do with all thesis-based writing. Include paraphrases, summaries, and quotations from the text (and outside sources, if you do research for the assignment). Body paragraphs support the topic sentences, which in turn support the thesis.
- Conclude by restating your thesis (using different words and an appropriate transition). Add a general statement about the work and its significance or, if applicable, its relation to culture, history, current events, art, or anything else outside it.
Use the applicable suggestions in Table 16.3 in planning your essay ( https://openstax.org/r/planning ):
| Title | |
| Introduction | Title(s), author(s), publication date(s), historical context, summary, thesis |
| Body paragraphs (as many as needed; minimum of 3) | Transition Claims and supporting evidence: your ideas, paraphrases and/or quotations from text |
| Counterclaim (if any) | Addressing opposing or negative ideas |
| Conclusion | Restatement of thesis, broader generalization |
Literary Nonfiction Model
A frequent theme in literary nonfiction is the examination of alternative ways of living, often solitary and away from society, and finding truth in individualism and self-sufficiency. Although most people live in social groups and willingly accept the identity and security that communities offer, dropping out and going it alone have long been a part of emotional as well as physical life for some.
You have the option to analyze the nonfiction accounts of writers exploring solitary human behavior in American life. If you select Another Lens 2 , you will read an excerpt from the story of Chris McCandless (1968–c. 1992), who chose a brief and uncomfortable solitary existence in Alaska. Or you can read the following section dealing with the works of Henry David Thoreau, the American philosopher and author who dropped out of society temporarily, largely because of his strong opposition to government policies he believed to be morally wrong and because of his refusal to conform to social practices and expectations he found objectionable.
Introduction
Figure \(16.7\) Henry David Thoreau, American naturalist, philosopher, and writer (credit: “Henry David Thoreau - Restored – greyscale” by Benjamin Maxham [1821–1899]/Wikimedia Commons, Public Domain)
Henry David Thoreau (1817–1862) is best known as a thinker and writer on nature, as reflected in his two famous works, the highly influential Civil Disobedience ( https://openstax.org/r/Civil-Disobedience ) (1849) and Walden ; or, Life in the Woods (1854). Both works celebrate individual freedoms: the right to protest against what one believes is morally or ethically wrong and the choice to live as one believes. In describing his life over a period of precisely two years, two months, and two days in a 10-by-15-foot cabin he built on Walden Pond, 20 miles northwest of Boston near Concord, Massachusetts, Thoreau wrote:
I went to the woods because I wished to live deliberately [carefully, unhurried], to front [confront] only the essential facts of life, and see if I could not learn what it had to teach, and not, when I came to die, discover that I had not lived. I did not wish to live what was not life, living is so dear; nor did I wish to practise resignation, unless it was quite necessary.
Thoreau’s insistence on standing by his principles and on living a simple life by choice are two abiding themes in his work. Even before the physical move to Walden, Thoreau had refused to pay his poll tax (granting him the right to vote) for a number of years because he strongly objected to the government’s use of his money to support enslavement and the war with Mexico. He went peacefully to jail as a result, until he was bailed out (the next day). In “Civil Disobedience,” Thoreau advocates for more individual freedom and for individuals to defy unjust laws in nonviolent ways. His writings on “passive resistance” inspired the thoughts and actions of influential figures such as Indian leader Mohandas Gandhi (1869–1948), American religious and civil rights leader Martin Luther King, Jr. (1929–1968), and other leaders of nonviolent liberation movements. In Walden , Thoreau describes and advocates for a simple life in which a person breaks with society when they feel the need to express their individualism, often based on ideas others do not share.
These themes are the focus of analysis in the following excerpts from an essay by student Alex Jones for a first-year composition class.
The Two Freedoms of Henry David Thoreau by Alex Jones
Henry David Thoreau led millions of people throughout the world to think of individual freedom in new ways. During his lifetime he attempted to live free of unjust governmental restraints as well as conventional social expectations. In his 1849 political essay “On the Duty of Civil Disobedience,” he makes his strongest case against governmental interference in the lives of citizens. In his 1854 book Walden ; or Life in the Woods , he makes the case for actually living free, as he did in his own life, from social conventions and expectations.
The title clearly identifies Thoreau and sets the expectation that two aspects or definitions of freedom will be discussed in two different works. Alex Jones wants readers to know that millions of people worldwide figure in Thoreau’s legacy. He gives the examples of “unjust governmental restraints” and “conventional social expectations" as the parts of social life. Thoreau rejected, this limiting the scope of the analysis and preparing for the body of the essay.
Jones notes the titles and publication dates of both works and immediately moves ahead to analyze the two works, "Civil Disobedience" first. He will show how this political statement leads to the narrative of Walden , the actual story of a man's life in temporary exile.
Thoreau opens “Civil Disobedience” with his statement “that government is best which governs not at all.” The analysis moves immediately to the first work to be discussed and features the memorable quotation regarding
The analysis moves immediately to the first work to be discussed and features the memorable quotation regarding a government that does not govern. The statement may seem contradictory, but for Thoreau it is a direct statement in that someone who allows himself to be imprisoned will find freedom by distancing himself from all others to prove his point.
He argues that a government should allow its people to be as free as possible while providing for their needs without interfering in daily life. In other words, in daily life a person attends to the business of eating, sleeping, and earning a living and not dealing in any noticeable way with an entity called “a government."
 The word government is repeated for emphasis as the negative subject of this essay—in literary terms, a powerful and constant antagonist that constrains and disempowers.
Because Thoreau did not want his freedom overshadowed by government regulations, he tried to ignore them. However, the American government of 1845 would not let him. He was arrested and put in the Concord jail for failing to pay his poll tax, a tax he believed unjust because it supported the government’s war with Mexico as well as the immoral institution of slavery. Instead of protesting his arrest, he celebrated it and explained its meaning by writing “Civil Disobedience,” one of the most famous English-language essays ever written. In it, he argues persuasively, “Under a government which imprisons any unjustly, the true place for a just man is also a prison” (230). Thus, the idea of passive resistance—and accepting unjust arrest to make a point—was formed, a doctrine that advocated protest against the government by nonviolent means:
How does it become a man to behave toward this American government today? I answer that he cannot without disgrace be associated with it. I cannot for an instant recognize that political organization as my government which is the slave’s government also. (224)
Jones strengthens his own writing by calling the essay one of the most famous works ever written. This is not an ordinary technique in textual analysis, but when done for emphasis, it helps the analysis gain power. Using “instead of protesting” at the start of his sentence is another example of strong contrast and linkage.
For nearly 200 years, Thoreau’s formulation of passive resistance has been a part of the human struggle for freedom. In fact, it changed the world by inspiring the resistance movements led by Mohandas Gandhi and Martin Luther King, Jr.
The total effect is to make Jones’s analytical essay more important for readers, as Thoreau’s writings have indeed changed the world despite being written humbly as the voice of one man’s conscience and isolation in his own freedom.
Thoreau also wanted to be free from the everyday pressures to conform to society’s expectations.
Jones transitions from the first short work to the different and equally famous nonfiction narrative Walden, moving smoothly from one freedom to the next with the transition “also wanted.” This second analysis of freedom is the second part of the essay’s thesis.
He believed in doing and possessing only the essential things in life. To demonstrate his case, in 1845, he moved to the outskirts of Concord, Massachusetts, and lived by himself for just over two years in a cabin he built at Walden Pond. Thoreau wrote Walden to explain the value of living simply, far removed from the unnecessary complexity of society: “simplicity, simplicity, simplicity! I say, let your affairs be as two or three, and not a hundred or a thousand” (66). At Walden, he lived as much as possible by this statement, building his own house and furniture, growing his own food, bartering for simple necessities, and attending to his own business rather than seeking employment from others.
Jones uses textual evidence to support his claim. He summarizes Thoreau’s activities at Walden and quotes Thoreau as evidence to reinforce the freedom of mind that simple living allows.
Figure \(16.6\) Setting for Thoreau’s cabin at Walden Pond near Concord (credit: “Site of Thoreau’s Hut, Concord, Mass“ by Detroit Publishing Company/Wikimedia Commons, Public Domain)
Living at Walden Pond gave Thoreau the chance to formulate many of his ideas about living an unencumbered, economical life. At Walden, he lived simply to “front only the essential facts of life” (66) and to center his thoughts on “living” instead of on unnecessary details of mere livelihood. He developed survival skills that freed him from the constraints of city dwellers whose lives depended upon a web of material things and services provided by others. He preferred to “take rank hold on life and spend my day more as animals do” (117).
Jones uses the poetic language of high rhetoric directly from Thoreau. The body of the essay gives specific evidence of how Thoreau ate, built, read, and provided for his needs, cutting away all but the essential man in the two settings of his life.
While living at Walden Pond, Thoreau was free to occupy his time in any way that pleased him, which for him meant mostly writing and tending his bean patch. The details of his gardening appear frequently, as he concentrated on it during his time there.  His neighbors accused him of being selfish and did not understand that he sought most of all “to live deliberately” (66), as he felt all people should learn to do.
Then, as now, most people had more responsibilities than Thoreau had and could not just pack up their belongings and go live in the woods—if they could find free woods to live in. Today, people are intrigued to read about Thoreau’s experiences and are inspired by his thoughts, but few people can actually live or do as he suggests. The idea of life without cell phones or Internet seems inconceivable, even if one grows one’s own food and lives mostly off the grid.
The next-to-last paragraph recognizes what could be a counterclaim: not everyone in contemporary times would view living alone for two years as a pleasure. Rather, they might see it as a different kind of prison, perhaps even a dangerous one. Indeed, such deprivation has less appeal these days, and people who do go off by themselves may be seen to have questionable motives.
The theme of exploring how a man lives in or outside governmental control is clear in the choices he must make to define himself as a free person. Nevertheless, practical or not, Thoreau’s writings about freedom from government and society have inspired countless people to reassess how they live their lives. Though unable to live as Thoreau advocated, readers everywhere remain inspired by his ideals and his belief in the two freedoms.
Jones concludes by emphasizing the strength of Thoreau’s ideas—his two freedoms—and the influence they have had in the world.
Review the Essay
After reading Alex Jones’s essay, complete the following sentences to review his work:
- He identifies and summarizes the content by ________.
- He describes the form and structure of Thoreau’s works when ________.
- He places Thoreau and his works in context by ________.
- He clearly states his own theme in reading Thoreau, which is ________.
- He indicates Thoreau’s unusual language at times, such as ________.
- He gives supporting evidence for his points, such as ________.
- He includes a visual to ________.
- He concludes with a balanced and convincing viewpoint by ________.
Figure \(16.9\) Contemporary replica of Thoreau’s cabin and statue of Thoreau (credit: “Walden pond - Thoreau’s house” by Luigi Franchini/flickr, CC BY 2.0)
For Reference: excerpt from Walden ( https://openstax.org/r/Civil-Disobedience ) by Henry David Thoreau from “Where I Lived and What I Lived For”
When first I took up my abode in the woods, that is, began to spend my nights as well as days there, which, by accident, was on Independence Day, or the Fourth of July, 1845, my house was not finished for winter, but was merely a defense against the rain, without plastering or chimney, the walls being of rough, weather-stained boards, with wide chinks, which made it cool at night. The upright white hewn studs and freshly planed door and window casings gave it a clean and airy look, especially in the morning, when its timbers were saturated with dew, so that I fancied that by noon some sweet gum would exude from them. To my imagination it retained throughout the day more or less of this auroral character, reminding me of a certain house on a mountain which I had visited the year before. This was an airy and unplastered cabin, fit to entertain a travelling god, and where a goddess might trail her garments. The winds which passed over my dwelling were such as sweep over the ridges of mountains, bearing the broken strains, or celestial parts only, of terrestrial music.…
The only house I had been the owner of before, if I except a boat, was a tent, which I used occasionally when making excursions in the summer, and this is still rolled up in my garret; but the boat, after passing from hand to hand, has gone down the stream of time. With this more substantial shelter about me, I had made some progress toward settling in the world. This frame, so slightly clad, was a sort of crystallization around me, and reacted on the builder. It was suggestive somewhat as a picture in outlines. I did not need to go outdoors to take the air, for the atmosphere within had lost none of its freshness. It was not so much within doors as behind a door where I sat, even in the rainiest weather. The Harivansa [important Sanskrit text] says, “An abode without birds is like a meat without seasoning.” Such was not my abode, for I found myself suddenly neighbor to the birds; not by having imprisoned one, but having caged myself near them.…
For the first week, whenever I looked out on the pond it impressed me like a tarn high up on the side of a mountain, its bottom far above the surface of other lakes, and, as the sun arose, I saw it throwing off its nightly clothing of mist, and here and there, by degrees, its soft ripples or its smooth reflecting surface was revealed, while the mists, like ghosts, were stealthily withdrawing in every direction into the woods, as at the breaking up of some nocturnal conventicle. The very dew seemed to hang upon the trees later into the day than usual, as on the sides of mountains.
This small lake was of most value as a neighbor in the intervals of a gentle rain storm in August, when, both air and water being perfectly still, but the sky overcast, mid-afternoon had all the serenity of evening, and the wood-thrush sang around, and was heard from shore to shore. A lake like this is never smoother than at such a time; and the clear portion of the air above it being shallow and darkened by clouds, the water, full of light and reflections, becomes a lower heaven itself so much the more important. From a hill top nearby, where the wood had been recently cut off, there was a pleasing vista southward across the pond, through a wide indentation in the hills which form the shore there, where their opposite sides sloping toward each other suggested a stream flowing out in that direction through a wooded valley, but stream there was none. That way I looked between and over the near green hills to some distant and higher ones in the horizon, tinged with blue. Indeed, by standing on tiptoe I could catch a glimpse of some of the peaks of the still bluer and more distant mountain ranges in the north-west, those true-blue coins from heaven’s own mint, and also of some portion of the village. But in other directions, even from this point, I could not see over or beyond the woods which surrounded me. It is well to have some water in your neighborhood, to give buoyancy to and float the earth. One value even of the smallest well is, that when you look into it you see that earth is not continent but insular. This is as important as that it keeps butter cool. When I looked across the pond from this peak toward the Sudbury meadows, which in time of flood I distinguished elevated perhaps by a mirage in their seething valley, like a coin in a basin, all the earth beyond the pond appeared like a thin crust insulated and floated even by this small sheet of interverting water, and I was reminded that this on which I dwelt was but dry land .
We must learn to reawaken and keep ourselves awake, not by mechanical aids, but by an infinite expectation of the dawn, which does not forsake us in our soundest sleep. I know of no more encouraging fact than the unquestionable ability of man to elevate his life by a conscious endeavor. It is something to be able to paint a particular picture, or to carve a statue, and so to make a few objects beautiful; but it is far more glorious to carve and paint the very atmosphere and medium through which we look, which morally we can do. To affect the quality of the day, that is the highest of arts. Every man is tasked to make his life, even in its details, worthy of the contemplation of his most elevated and critical hour. If we refused, or rather used up, such paltry information as we get, the oracles would distinctly inform us how this might be done.
I went to the woods because I wished to live deliberately, to front only the essential facts of life, and see if I could not learn what it had to teach, and not, when I came to die, discover that I had not lived. I did not wish to live what was not life, living is so dear; nor did I wish to practise resignation, unless it was quite necessary. I wanted to live deep and suck out all the marrow of life, to live so sturdily and Spartan-like as to put to rout all that was not life, to cut a broad swath and shave close, to drive life into a corner, and reduce it to its lowest terms, and, if it proved to be mean, why then to get the whole and genuine meanness of it, and publish its meanness to the world; or if it were sublime, to know it by experience, and be able to give a true account of it in my next excursion. For most men, it appears to me, are in a strange uncertainty about it, whether it is of the devil or of God, and have somewhat hastily concluded that it is the chief end of man here to “glorify God and enjoy him forever.”
Still we live meanly, like ants; though the fable tells us that we were long ago changed into men; like pygmies we fight with cranes; it is error upon error, and clout upon clout, and our best virtue has for its occasion a superfluous and evitable wretchedness. Our life is frittered away by detail. An honest man has hardly need to count more than his ten fingers, or in extreme cases he may add his ten toes, and lump the rest. Simplicity, simplicity, simplicity! I say, let your affairs be as two or three, and not a hundred or a thousand; instead of a million count half a dozen, and keep your accounts on your thumb nail. In the midst of this chopping sea of civilized life, such are the clouds and storms and quicksands and thousand-and-one items to be allowed for, that a man has to live, if he would not founder and go to the bottom and not make his port at all, by dead reckoning, and he must be a great calculator indeed who succeeds. Simplify, simplify. Instead of three meals a day, if it be necessary eat but one; instead of a hundred dishes…
Why should we live with such… waste of life? We are determined to be starved before we are hungry. Men say that a stitch in time saves nine, and so they take a thousand stitches to-day to save nine to-morrow.
Peer Review:
After you have completed your first draft, exchange essays with a partner for peer review. Look at the questions you answered to address the essay about Thoreau. Then, to provide helpful feedback, answer these questions about your peer’s draft.
- Does the introduction include the author, title of the work, publication date, historical context, and a brief summary?
- What is your peer’s main claim, or thesis? Is it clearly stated? If not, how might your peer clarify it?
- Is the thesis effectively supported throughout the essay? How does each paragraph support the thesis? What evidence does each contain? Has the writer included direct quotations, paraphrases, and summary as relevant and convincing support? Is there enough information to sustain the writer’s claims? How might the author improve their support? In working on this section, go through each body paragraph separately for these criteria.
- Does the analysis address counterclaims? If not, how might the writer include them?
- Which sentence or sentences restate the thesis? If a restatement is not there, what might the writer include?
Revising:
Once you have feedback from a peer, consider their suggestions. Read all comments, and think carefully before making changes.
- Use your discretion. Sometimes writers do not agree with their peers’ suggestions; indeed, authors do not always revise everything suggested by editors. However, it is important to clarify what might have prompted a response from a peer, such as “This seems like more of an unsupported opinion than text-based evidence.” Here you might consider including a source citation either from the text or from an outside resource, or consider further explaining your claim. However, if you think your peer reviewer misinterpreted or read your claim superficially, do not revise it. At all times, though, maintain ownership. It is your paper; you are the ultimate judge of whether the ideas in it represent you and your views. Never include someone else’s idea in your paper if you do not understand it or believe it. Whether or not you decide to revise, be sure to read and consider all suggestions carefully.
- Focus on global suggestions first. Global feedback applies to your entire paper. You may have to revise your topic or thesis so that your paper meets assigned guidelines or does what it should. It is important to revise global feedback first, for these revisions might necessitate changes in content and organization, among other things.
- Complete a close revision. Check your paper to revise for clarity at the sentence level, and double-check citations, if you have them, for accuracy and style.
Student Revision Model
Below is a paragraph from the first draft of Gwyn Garrison’s paper. It was reviewed by a peer, who made the suggestions indicated. First, read the draft. Next, read the reviewer’s suggestions and consider whether you would accept or reject each one. Then, read the paragraph as it appears in the final version. After each suggestion, consider why you think Gwyn Garrison accepted or rejected the reviewer’s comment.
Original Draft
When Calixta acts outside of societal norms, she discovers the freedom of self-expression and passion. Chopin’s diction evokes a spiritual transcendence that allows Calixta to exist momentarily outside social norms that exist only in the physical plane of existence: “when he possessed her, they seemed to swoon at the very borderland of life’s mystery.” The affair becomes a vehicle that allows Calixta to get to a place of true selfexpression. The storm, an aspect of nature or the natural world, acts as the catalyst in Calixta’s natural selfrealization of womanhood. The storm breaks externally and internally for Calixta. Chopin’s depiction of Calixta’s sexual liberation and fulfillment outside of her marriage is an early step in the fight to bridge the gap between women’s bodies and their sociopolitical lives. By presenting female sexuality in a way that is enlightening rather than degrading, Chopin helps destigmatize labels such as whore , which have been used to shame women for acting outside of traditional gender expectations.
Peer Reviewer's Comments
- A transition would help link this paragraph with the previous one.
- At the beginning of the paragraph, after the first sentence, add a short description or explanation of what is happening in the scene.
- The quotation from the text doesn’t help explain your claim. Anyway, you left out a word.
- Perhaps you could add a quotation about the storm.
- Can you clarify the relationship between the storm and Calixta’s self-realization?
Final Version
When Calixta acts outside of societal norms, however , she discovers the freedom of self-expression and passion. All of the parts of her womanhood that have no place in the society in which she lives have been repressed until this one moment. In this scene, Chopin takes possession of the term whore and redefines Calixta’s behavior as a transformative awakening. Chopin’s diction evokes a spiritual transcendence that allows Calixta to exist momentarily outside social norms that exist only in the physical plane of existence: “when he possessed her, they seemed to swoon together at the very borderland of life’s mystery.” The affair becomes a vehicle that allows Calixta to get to a place of true self-expression. The storm, an aspect of nature or the natural world, acts as the catalyst in Calixta’s natural self-realization of womanhood. As the storm breaks externally, it also breaks internally for Calixta. Chopin’s depiction of Calixta’s sexual liberation and fulfillment outside of her marriage is an early step in the fight to bridge the gap between women’s bodies and their sociopolitical lives. By presenting female sexuality in a way that is enlightening rather than degrading, Chopin helps destigmatize labels such as whore , which have been used to shame women for acting outside of traditional gender expectations.


13.8: Review Questions
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Which disease-related alteration can potentially place an older client with chronic obstructive pulmonary disease (COPD) at risk for altered nutrition?
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Altered taste
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Decreased energy requirements
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Fatigue
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Altered smell
On initial laboratory tests, an adult client’s albumin level is below 3.5 g/dL. Which additional test could potentially be utilized to gather additional information about nutritional risk for malnutrition?
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Hemoglobin A1C
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Antinuclear antibodies (ANA)
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Prealbumin
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Chest x-ray
What is the process for carbon dioxide and oxygen exchange?
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Ventilation
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Gas exchange
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Perfusion
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Diffusion
The nurse is instructing an older client on the importance of adequate calcium and vitamin D intake. Which of the following age-related changes would this intervention affect?
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Decreased lung capacity
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Increased muscle atrophy
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Decreased elasticity
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Increased vascular resistance
For the client with chronic obstructive pulmonary disease (COPD), which dietary teaching should be incorporated?
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High-fat food choices
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Simple carbohydrates as primary food choice
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High-protein, calorie-dense food choices
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Low-fiber food choices
Which of the following is a primary goal of nutrition therapy for a client with chronic obstructive pulmonary disease (COPD)?
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Maintaining a healthy body weight
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Improving oxygenation status
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Preventing muscle wasting
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Lowering CO 2 levels
The nurse is instructing a client with chronic obstructive pulmonary disease (COPD) on strategies to maximize food intake. Which of the following responses by the client indicates additional education is needed?
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“I will treat myself to cake or pudding for dessert.”
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“I will eat add cheese to my food whenever possible.”
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“I will eat more cold foods rather than hot meals.”
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“I will try to eat as much as I can during the evening before I go to bed.”
A client with chronic obstructive pulmonary disease (COPD) is being assessed by a nurse. With palpitation, the nurse feels crackling. What does the crackling most likely indicate?
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Fremitus
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Orthopnea
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Crepitus
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Tuberculosis
Which of the following findings is the best indicator that the client has followed the prescribed nutritional plan to maintain a healthy weight?
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The client can verbalize why obesity is detrimental to pulmonary health.
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The client can describe the MyPlate approach to food selection.
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The client’s food log demonstrates adherence to healthy nutritional plan.
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The client has reached their recommended weight range.
A client states that they are experiencing a nighttime cough. Which medication could explain this symptom?
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Acetaminophen
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Ibuprofen
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Beta-2 agonist
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Angiotensin-converting enzyme (ACE) inhibitors