3.4: Composition of Functions
Learning Objectives
- Combine functions using algebraic operations.
- Create a new function by composition of functions.
- Evaluate composite functions.
- Find the domain of a composite function.
- Decompose a composite function into its component functions.
Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.
Using descriptive variables, we can notate these two functions. The function \(C(T)\) gives the cost \(C\) of heating a house for a given average daily temperature in \(T\) degrees Celsius. The function \(T(d)\) gives the average daily temperature on day d of the year. For any given day, \(Cost=C(T(d))\) means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature \(T(d)\). For example, we could evaluate \(T(5)\) to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the cost function at that temperature. We would write \(C(T(5))\).
By combining these two relationships into one function, we have performed function composition, which is the focus of this section.
Combining Functions Using Algebraic Operations
Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.
Suppose we need to add two columns of numbers that represent a husband and wife’s separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that year’s incomes and then collecting all the data in a new column. If \(w(y)\) is the wife’s income and \(h(y)\) is the husband’s income in year \(y\), and we want \(T\) to represent the total income, then we can define a new function.
\[T(y)=h(y)+w(y) \nonumber\]
If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write
\[T=h+w \nonumber\]
Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.
For two functions \(f(x)\) and \(g(x)\) with real number outputs, we define new functions \(f+g\), \(f−g\), \(fg\), and \(\frac{f}{g}\) by the relations.
\[ \begin{align*} (f+g)(x) &=f(x)+g(x) \\[4pt] (f−g)(x) &=f(x)−g(x) \\[4pt] (fg)(x)&=f(x)g(x) \\[4pt] \left(\dfrac{f}{g}\right)(x) &=\dfrac{f(x)}{g(x)} \end{align*}\]
Example \(\PageIndex{1}\): Performing Algebraic Operations on Functions
Find and simplify the functions \((g−f)(x)\) and \(\left(\dfrac{g}{f}\right)(x)\), given \(f(x)=x−1\) and \(g(x)=x^2−1\). Are they the same function?
Solution
Begin by writing the general form, and then substitute the given functions.
\[\begin{align*} (g−f)(x) &= g(x)−f(x) \\[4pt] (g−f)(x) &=x^2−1−(x−1) \\[4pt] &=x^2−x \\[4pt] &=x(x−1) \end{align*}\]
\[\begin{align*} \left(\dfrac{g}{f}\right)(x)&=g(x)f(x) \\[4pt] \left(\dfrac{g}{f}\right)(x)&=\dfrac{x^2−1}{x−1} \\[4pt] &=\dfrac{(x+1)(x−1)}{x−1} \\[4pt]&=x+1 \end{align*}\]
No, the functions are not the same.
Note: For \(\left(\dfrac{g}{f}\right)(x)\), the condition \(x\neq1\) is necessary because when \(x=1\), the denominator is equal to 0, which makes the function undefined.
Exercise \(\PageIndex{1}\)
Find and simplify the functions \((fg)(x)\) and \((f−g)(x)\).
\[f(x)=x−1 \nonumber\]
and
\[g(x)=x^2−1 \nonumber\]
Are they the same function?
- Answer
-
\((fg)(x)=f(x)g(x)=(x−1)(x2−1)=x^3−x^2−x+1 \\[4pt] (f−g)(x)=f(x)−g(x)=(x−1)−(x^2−1)=x−x^2\)
No, the functions are not the same.
Create a Function by Composition of Functions
Performing algebraic operations on functions combines them into a new function, but we can also create functions by composing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output. The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions . The resulting function is known as a composite function . We represent this combination by the following notation:
\[f{\circ}g(x)=f(g(x))\]
We read the left-hand side as“\(f\) composed with \(g\) at \(x\),” and the right-hand side as“\(f\) of \(g\) of \(x\).”The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol \(\circ\) is called the composition operator. We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important not to confuse function composition with multiplication because, as we learned above, in most cases \(f(g(x)){\neq}f(x)g(x)\).
It is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the equation above, the function \(g\) takes the input \(x\) first and yields an output \(g(x)\). Then the function \(f\) takes \(g(x)\) as an input and yields an output \(f(g(x))\).
In general, \(f{\circ}g\) and \(g{\circ}f\) are different functions. In other words, in many cases \(f(g(x)){\neq}g(f(x))\) for all \(x\). We will also see that sometimes two functions can be composed only in one specific order.
For example, if \(f(x)=x^2\) and \(g(x)=x+2\), then
\[\begin{align*} f(g(x))&= f(x+2) \\[4pt]&=(x+2)^2 \\[4pt] &=x^2+4x+4 \end{align*}\]
but
\[\begin{align*} g(f(x))&= g(x^2) \\[4pt]&=x^2+2 \end{align*}\]
These expressions are not equal for all values of x, so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value \(x=−\frac{1}{2}\).
Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.
Composition of Functions
When the output of one function is used as the input of another, we call the entire operation a composition of functions. For any input \(x\) and functions \(f\) and \(g\), this action defines a composite function , which we write as \(f{\circ}g\) such that
\[(f{\circ}g)(x)=f(g(x))\]
The domain of the composite function \(f{\circ}g\) is all \(x\) such that \(x\) is in the domain of \(g\) and \(g(x)\) is in the domain of \(f\).
It is important to realize that the product of functions \(fg\) is not the same as the function composition \(f(g(x))\), because, in general, \(f(x)g(x){\neq}f(g(x))\).
Example \(\PageIndex{2}\): Determining whether Composition of Functions is Commutative
Using the functions provided, find \(f(g(x))\) and \(g(f(x))\). Determine whether the composition of the functions is commutative .
\[f(x)=2x+1 \;\;\;\; g(x)=3−x \nonumber\]
Solution
Let’s begin by substituting \(g(x)\) into \(f(x)\).
\[\begin{align*} f(g(x))&= 2(3−x)+1 \\[4pt] &=6−2x+1 \\[4pt] &=7−2x \end{align*}\]
Now we can substitute \(f(x)\) into \(g(x)\).
\[\begin{align*} g(f(x))&= 3−(2x+1) \\[4pt]&=3−2x−1 \\[4pt] &=2-2x \end{align*}\]
We find that \(g(f(x)){\neq}f(g(x))\), so the operation of function composition is not commutative.
Example \(\PageIndex{3}\): Interpreting Composite Functions
The function \(c(s)\) gives the number of calories burned completing \(s\) sit-ups, and \(s(t)\) gives the number of sit-ups a person can complete in \(t\) minutes. Interpret \(c(s(3))\).
Solution
The inside expression in the composition is \(s(3)\). Because the input to the \(s\)-function is time, \(t=3\) represents 3 minutes, and \(s(3)\) is the number of sit-ups completed in 3 minutes.
Using \(s(3)\) as the input to the function \(c(s)\) gives us the number of calories burned during the number of sit-ups that can be completed in 3 minutes, or simply the number of calories burned in 3 minutes (by doing sit-ups).
Example \(\PageIndex{4}\): Investigating the Order of Function Composition
Suppose \(f(x)\) gives miles that can be driven in \(x\) hours and \(g(y)\) gives the gallons of gas used in driving \(y\) miles. Which of these expressions is meaningful: \(f(g(y))\) or \(g(f(x))\)?
Solution
The function \(y=f(x)\) is a function whose output is the number of miles driven corresponding to the number of hours driven.
\[\text{number of miles } =f (\text{number of hours}) \nonumber\]
The function \(g(y)\) is a function whose output is the number of gallons used corresponding to the number of miles driven. This means:
\[\text{number of gallons } =g(\text{number of miles}) \nonumber\]
The expression \(g(y)\) takes miles as the input and a number of gallons as the output. The function \(f(x)\) requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression \(f(g(y))\) is meaningless.
The expression \(f(x)\) takes hours as input and a number of miles driven as the output. The function \(g(y)\) requires a number of miles as the input. Using \(f(x)\) (miles driven) as an input value for \(g(y)\), where gallons of gas depends on miles driven, does make sense. The expression \(g(f(x))\) makes sense, and will yield the number of gallons of gas used, \(g\), driving a certain number of miles, \(f(x)\), in \(x\) hours.
Question/Answer
Are there any situations where \(f(g(y))\) and \(g(f(x))\) would both be meaningful or useful expressions?
Yes. For many pure mathematical functions, both compositions make sense, even though they usually produce different new functions. In real-world problems, functions whose inputs and outputs have the same units also may give compositions that are meaningful in either order
Exercise \(\PageIndex{2}\)
The gravitational force on a planet a distance \(r\) from the sun is given by the function \(G(r)\). The acceleration of a planet subjected to any force \(F\) is given by the function \(a(F)\). Form a meaningful composition of these two functions, and explain what it means.
- Answer
-
A gravitational force is still a force, so \(a(G(r))\) makes sense as the acceleration of a planet at a distance \(r\) from the Sun (due to gravity), but \(G(a(F))\) does not make sense.
Evaluating Composite Functions
Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner function’s output as the input for the outer function.
Evaluating Composite Functions Using Tables
When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.
Example \(\PageIndex{5}\): Using a Table to Evaluate a Composite Function
Using Table \(\PageIndex{1}\), evaluate \(f(g(3))\) and \(g(f(3))\).
| \(x\) | \(f(x)\) | \(g(x)\) |
|---|---|---|
| 1 | 6 | 3 |
| 2 | 8 | 5 |
| 3 | 3 | 2 |
| 4 | 1 | 7 |
Solution
To evaluate \(f(g(3))\), we start from the inside with the input value 3. We then evaluate the inside expression \(g(3)\) using the table that defines the function \(g: g(3)=2\). We can then use that result as the input to the function \(f\), so \(g(3)\) is replaced by 2 and we get \(f(2)\). Then, using the table that defines the function \(f\), we find that \(f(2)=8\).
\[g(3)=2 \nonumber\]
\[f(g(3))=f(2)=8 \nonumber\]
To evaluate \(g(f(3))\), we first evaluate the inside expression \(f(3)\) using the first table: \(f(3)=3\). Then, using the table for \(g\), we can evaluate
\[g(f(3))=g(3)=2 \nonumber\]
Table \(\PageIndex{2}\) shows the composite functions \(f{\circ}g\) and \(g{\circ}f\) as tables.
| \(x\) | \(g(x)\) | \(f(g(x))\) | \(f(x)\) | \(g(f(x))\) |
|---|---|---|---|---|
| 3 | 2 | 8 | 3 | 2 |
Exercise \(\PageIndex{3}\)
Using Table \(\PageIndex{1}\), evaluate \(f(g(1))\) and \(g(f(4))\).
- Answer
-
\(f(g(1))=f(3)=3\) and \(g(f(4))=g(1)=3\)
Evaluating Composite Functions Using Graphs
When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the x- and y-axes of the graphs.
How To ...
Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs.
- Locate the given input to the inner function on the x-axis of its graph.
- Read off the output of the inner function from the y-axis of its graph.
- Locate the inner function output on the x-axis of the graph of the outer function.
- Read the output of the outer function from the y-axis of its graph. This is the output of the composite function.
Example \(\PageIndex{6}\): Using a Graph to Evaluate a Composite Function
Using Figure \(\PageIndex{3}\), evaluate \(f(g(1))\).
Solution
To evaluate \(f(g(1))\), we start with the inside evaluation. See Figure \(\PageIndex{4}\).
We evaluate \(g(1)\) using the graph of \(g(x)\), finding the input of 1 on the x-axis and finding the output value of the graph at that input. Here, \(g(1)=3\). We use this value as the input to the function \(f\).
\[f(g(1))=f(3) \nonumber\]
We can then evaluate the composite function by looking to the graph of \(f(x)\), finding the input of 3 on the x-axis and reading the output value of the graph at this input. Here, \(f(3)=6\), so \(f(g(1))=6\).
Analysis
Figure \(\PageIndex{5}\) shows how we can mark the graphs with arrows to trace the path from the input value to the output value.
Exercise \(\PageIndex{4}\)
Using Figure \(\PageIndex{3}\), evaluate \(g(f(2))\).
- Answer
-
\(g(f(2))=g(5)=3\)
Evaluating Composite Functions Using Formulas
When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.
While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition \(f(g(x))\). To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like \(f(t)=t^2−t\), we substitute the value inside the parentheses into the formula wherever we see the input variable.
How To...
Given a formula for a composite function, evaluate the function.
- Evaluate the inside function using the input value or variable provided.
- Use the resulting output as the input to the outside function.
Example \(\PageIndex{7}\): Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input
Given \(f(t)=t^2−t\) and \(h(x)=3x+2\), evaluate \(f(h(1))\).
Solution
Because the inside expression is \(h(1)\), we start by evaluating \(h(x)\) at 1.
\[ \begin{align*} h(1)=3(1)+2 \\[4pt] h(1)&=5 \end{align*} \]
Then \(f(h(1))=f(5)\), so we evaluate \(f(t)\) at an input of 5.
\[ \begin{align*} f(h(1)) &=f(5) \\[5pt] f(h(1))&=5^2−5 \\[5pt] f(h(1))&=20 \end{align*} \]
Analysis
It makes no difference what the input variables \(t\) and \(x\) were called in this problem because we evaluated for specific numerical values.
Exercise \(\PageIndex{5}\)
Given \(f(t)=t^2−t\) and \(h(x)=3x+2\), evaluate
a. \(h(f(2))\)
b. \(h(f(−2))\)
- Answer a
-
8
- Answer b
-
20
Finding the Domain of a Composite Function
As we discussed previously, the domain of a composite function such as \(f{\circ}g\) is dependent on the domain of \(g\) and the domain of \(f\). It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as \(f{\circ}g\). Let us assume we know the domains of the functions \(f\) and \(g\) separately. If we write the composite function for an input \(x\) as \(f(g(x))\), we can see right away that \(x\) must be a member of the domain of g in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that \(g(x)\) must be a member of the domain of \(f\), otherwise the second function evaluation in \(f(g(x))\) cannot be completed, and the expression is still undefined. Thus the domain of \(f{\circ}g\) consists of only those inputs in the domain of \(g\) that produce outputs from \(g\) belonging to the domain of \(f\). Note that the domain of \(f\) composed with \(g\) is the set of all \(x\) such that \(x\) is in the domain of \(g\) and g(x)\) is in the domain of \(f\).
Definition: Domain of a Composite Function
The domain of a composite function \(f(g(x))\) is the set of those inputs \(x\) in the domain of \(g\) for which \(g(x)\) is in the domain of \(f\).
How To...
Given a function composition \(f(g(x))\), determine its domain.
- Find the domain of \(g\).
- Find the domain of \(f\).
- Find those inputs \(x\) in the domain of \(g\) for which \(g(x)\) is in the domain of \(f\). That is, exclude those inputs \(x\) from the domain of \(g\) for which \(g(x)\) is not in the domain of \(f\). The resulting set is the domain of \(f{\circ}g\).
Example \(\PageIndex{8A}\): Finding the Domain of a Composite Function
Find the domain of
\[(f∘g)(x) \text{ where } f(x)=\dfrac{5}{x−1} \text{ and } g(x)=\dfrac{4}{3x−2} \nonumber\]
Solution
The domain of \(g(x)\) consists of all real numbers except \(x=\frac{2}{3}\), since that input value would cause us to divide by 0. Likewise, the domain of \(f\) consists of all real numbers except 1. So we need to exclude from the domain of \(g(x)\) that value of \(x\) for which \(g(x)=1\).
\[\begin{align*} \dfrac{4}{3x-2}&= 1 \\[4pt] 4 &=3x-2 \\[4pt] 6&=3x \\[4pt] x&= 2 \end{align*}\]
So the domain of \(f{\circ}g\) is the set of all real numbers except \(\frac{2}{3}\) and \(2\). This means that
\[x{\neq} \dfrac{2}{3} \text{ or } x\neq2 \nonumber\]
We can write this in interval notation as
\[\left(−\infty,\dfrac{2}{3}\right)\cup \left(\dfrac{2}{3},2 \right)\cup \left(2,\infty \right) \nonumber\]
Example \(\PageIndex{8B}\): Finding the Domain of a Composite Function Involving Radicals
Find the domain of
\[(f{\circ}g)(x) \text{ where } f(x)=\sqrt{x+2} \text{ and } g(x)=\sqrt{3−x} \nonumber\]
Solution
Because we cannot take the square root of a negative number, the domain of \(g\) is \(\left(−\infty,3\right]\). Now we check the domain of the composite function
\[(f{\circ}g)(x)=\sqrt{\sqrt{3−x}+2} \nonumber\]
For \((f∘g)(x)=\sqrt{ \sqrt{3−x}+2},\sqrt{3−x}+2≥0,\) since the radicand of a square root must be positive. Since square roots are positive, \(\sqrt{3−x}≥0\),or, \(3−x≥0,\) which gives a domain of \((-∞,3]\).
Analysis
This example shows that knowledge of the range of functions (specifically the inner function) can also be helpful in finding the domain of a composite function. It also shows that the domain of \(f{\circ}g\) can contain values that are not in the domain of \(f\), though they must be in the domain of \(g\).
Exercise \(\PageIndex{6}\)
Find the domain of
\[(f{\circ}g)(x) \text{ where } f(x)=\dfrac{1}{x−2} \text{ and } g(x)=\sqrt{x+4} \nonumber\]
- Answer
-
\([−4,0)∪(0,∞)\)
Decomposing a Composite Function into its Component Functions
In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function , so we may choose the decomposition that appears to be most expedient.
Example \(\PageIndex{9}\): Decomposing a Function
Write \(f(x)=\sqrt{5−x^2}\) as the composition of two functions.
Solution
We are looking for two functions, \(g\) and \(h\), so \(f(x)=g(h(x))\). To do this, we look for a function inside a function in the formula for \(f(x)\). As one possibility, we might notice that the expression \(5−x^2\) is the inside of the square root. We could then decompose the function as
\[h(x)=5−x^2 \text{ and } g(x)=\sqrt{x} \nonumber\]
We can check our answer by recomposing the functions.
\[g(h(x))=g(5−x^2)=\sqrt{5−x^2} \nonumber\]
Exercise \(\PageIndex{7}\)
Write \(f(x)=\dfrac{4}{3−\sqrt{4+x^2}}\) as the composition of two functions.
- Answer
-
Possible answers:
\(g(x)=\sqrt{4+x^2}\)
\(h(x)=\dfrac{4}{3−x}\)
\(f=h{\circ}g\)
Access these online resources for additional instruction and practice with composite functions.
- Composite Functions ( http://openstaxcollege.org/l/compfunction )
- Composite Function Notation Application ( http://openstaxcollege.org/l/compfuncnot )
- Composite Functions Using Graphs ( http://openstaxcollege.org/l/compfuncgraph )
- Decompose Functions ( http://openstaxcollege.org/l/decompfunction )
- Composite Function Values ( http://openstaxcollege.org/l/compfuncvalue )
Key Equation
- Composite function \((f{\circ}g)(x)=f(g(x))\)
Key Concepts
- We can perform algebraic operations on functions. See Example.
- When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.
- The function produced by combining two functions is a composite function. See Example and Example.
- The order of function composition must be considered when interpreting the meaning of composite functions. See Example.
- A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
- A composite function can be evaluated from a table. See Example.
- A composite function can be evaluated from a graph. See Example.
- A composite function can be evaluated from a formula. See Example.
- The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See Example and Example.
- Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
- Functions can often be decomposed in more than one way. See Example.
Glossary
- composite function
-
the new function formed by function composition, when the output of one function is used as the input of another


1: AGENDA
Organization of the CHEM 210 Course
The title of this course "Modern Applications of Chemistry" embraces the fact that there are essential underlying principles that can be found in what seem to be radically complicated systems. Much of Modern Chemistry appears to be quite different from what you have seen in prior chemistry courses, but at the end of this course you should begin to see how even complicated systems can be interpreted through essentially simple concepts and models.
This course is assessed in two main ways: Formative and Summative assessments. Formative Assessments are meant to be a place where you receive detailed feedback to help you to improve and to master certain critical ideas - things we call " Core Goals " in this course. There are 7 Core Goals listed below.
Additionally, There will be two midterms and one final exam that will focus on approximately 1/3 of the material each. It is important to note that the last third of the material is on applications of the prior 2/3 of the course in new situations, so it very much still will seem to be cumulative. These summative assessments will evaluate your ability to demonstrate your ability on longer more complicated problems, as well as on basic Learning Objectives.
Formative Assessments for Demonstrating Skill on Core Goals:
Instead of a Homework or Quiz grade that gives points based on effort or ability, you will be given multiple tries to try to pass each of the Core Goals above. Your work will be graded on a rubric and be given either a mark of "M" for Meets Expectations or an "N" for Not yet. The schedule for these assessments is below:
| Core Goal Attempt Date | First Attempt | Second Attempt | Third Attempt | Duration |
|---|---|---|---|---|
| Thur: 5/18/2023 | ( MAT ) | — | — | 30 min |
| Tue: 5/23/2023 | MAT | — | — | 20 min |
| Wed: 5/24/2023 | SOL | MAT | — | 30 min |
| Tue: 5/30/2023 | AB1 | SOL | — | 30 min |
|
Midterm I
Thur: 6/1/2023 |
— | — | MAT / SOL | 30 min |
| Tue: 6/6/2023 | AB2 | AB1 | — | 30 min |
| Wed: 6/7/2023 | ECHEM | AB2 | — | 30 min |
| Tue: 6/13/2023 | — | ECHEM | — | 20 min |
|
Midterm II
Thur: 6/15/2023 |
— | — | AB1 / AB2 / ECHEM | 45 min |
| Tue: 6/20/2023 | TMC | — | — | 20 min |
| Wed: 6/21/2023 | APP | TMC | — | 40 min |
|
Final Exam
Wed: 6/28/2023 |
— | APP | TMC | 40 min |
Note that only two attempts at the APP Core Goal attempts are given as these are cumulative applications of other materials. They are slightly longer than the others and time has been allocated for those. Note that the APP Core Goal is the 7th of the 7 goals, and is really intended to distinguish between an A- ability and an A ability in the course demonstrating an in-depth understanding of the applications of the other 6 Core Goals.
You only need to earn an "M" on a Core Goal ONCE. Once you have received a mark of "M" then you do not need to study for, nor take the subsequent attempts at that Core Goal!
Your grade on the Core Goals will be determined by the following Rubric:
- Earning an "M" on all 7 Core Goals: 100%
- 6 out of 7: 95%
- 5 out of 7: 80%
- 4 out of 7: 65%
- 3 out of 7: 50%
- 2 out of 7: 35%
- 1 out of 7: 20%
- 0 out of 7: 0%
Summative Assessments: Midterms and Final
The course has three summative assessments: Midterm I, Midterm II, and the Final Exam. Note that the Final Exam is the same duration as the standard exams (~1.5 hours), and just like the other exams, will have the time allotted at the end for Core Goal Assessments (recall that if you already met the Core Goal with an "M", then you will not need to stay to complete an additoinal attempt).
| Midterm I: Matter and Interactions | Midterm II: Solutions and Electrochemistry | Final Exam: Applications of Modern Chemistry |
|---|---|---|
|
|
|
Supplemental Resources:
Table of Thermodynamic Properites that is a little longer than the LibreTexts one: https://www.chm.uri.edu/weuler/chm112/refmater/thermtable.html
Core Goal MAT: States of Matter, Interactions, Bonding, and Energy
-
To gain a fundamental understanding of the composition of atoms with an emphasis on the atom’s nucleus.
-
To develop chemical literacy regarding isotopes, nuclear stability, radioactive decay, fission, fusion, and applications involving nuclear transformation.
- To develop chemical literacy regarding phases of matter including gases, liquids, and solids with an emphasis on:
- the gas phase and the origins and relative strengths of different contributions to intermolecular forces in the condensed phases,
- phase changes and phase diagrams,
- the properties of liquids and especially how they reflect intermolecular forces,
- the properties of solids, especially crystal structures and metallic conduction.
- To gain a fundamental understanding of the gaseous state of matter and the ideal and non-ideal behavior of gases.
| Lectures | Readings | Assessments |
|---|---|---|
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Assumed Knowledge from the Primer
Core Goal SOL: Solutions
-
To develop chemical literacy regarding the properties of solutions, with an emphasis on
- the miscibility of liquids and the solubility of gases and solids in liquids,
- the vapor pressure of solutions,
- colligative properties, and
- how these properties reflect the underlying intermolecular forces and thermodynamics.
-
To develop chemical literacy regarding chemical equilibrium as it applies to:
- solubility of ionic compounds and
- the formation of complex ions.
The SOL Core Goal Skill:
It is the expectation that you should be able to relate \(\Delta G\), \(\Delta S\), and \(\Delta H\) for a system, determine whether it is spontaneous and whether it is for enthalpic or entropic driven reasons. Then: you should be able to use the relationship between \(K\) and \(\Delta G^\circ\) to relate this to an equilibrium process.
Finally: you should be able to determine equilibrium concentrations or pressures from a given \(K\) (or a calculated \(K\)).
| Lectures | Readings | Assessments |
|---|---|---|
|
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Assumed Knowledge from the Primer
Core Goal AB1: Equilibrium Acids and Bases
Demonstrate chemical literacy regarding chemical equilibrium with an emphasis on:
- how chemical systems change as they move toward a state of equilibrium;
- how the equilibrium condition itself can be characterized;
- how a system at equilibrium will respond to changes in conditions;
- the relationship chemical equilibrium to chemical kinetics and thermodynamics;
- the relative reactivity of acids and bases;
- quantitative calculations involving chemical equilibria & aqueous solutions of acids, bases, and salts.
The AB1 Core Goal Skill:
Given any concentration of any strong acid, strong base, weak acids, weak bases, or a salt with either an acidic cation or a basic anion, be able to determine the pH and any of the other requested quantities including pOH, \(K_a\), \(K_b\), p\(K_a\), p\(K_b\) and/or percent ionization.
| Lectures | Readings | Assessments |
|---|---|---|
|
|
Assumed Knowledge from the Primer
Core Goal AB2: Quantitative Properties of Acid/Base Systems
To develop chemical literacy regarding mixtures of acids, bases, and buffers with an emphasis on the principles of chemical equilibrium as applied to (1) neutralization reactions, (2) buffered solutions, and (3) acid-base titrations.
The AB2 Core Goal Skill:
Given a known concentration and volume of a diprotic weak acid or base, and a known concentration of the opposite strong base/acid, be able to draw a titration curve including: (1) Initial pH, (2) first midpoint, (3) first equivalence point, (4) second midpoint, (5) second equivalence point, (6) strong titrant dilution (XS).
| Lectures | Readings | Assessments |
|---|---|---|
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Core Goal ECHEM: Electrochemistry and RedOx Chemistry
To develop chemical literacy regarding electron transfer reactions and their applications with an emphasis on (1) galvanic, electrolytic, and concentration cells, and (2) cell potentials and their relationship to chemical equilibrium and thermodynamics.
The ECHEM Core Goal Skill:
Be able to draw a galvanic or electrolytic cell, label it, and determine the spontaneous voltage (galvanic) or needed voltage (electrolytic) to get the cell to run.
| Lectures | Readings | Assessments |
|---|---|---|
|
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Core Goals TMC: Transition Metals and their Complexes
To develop chemical literacy regarding transition metals and coordination chemistry with emphasis on: (1) formation of and bonding in coordination compounds and the characterization of their isomers; (2) electronic structure and properties (magnetic, optical) of transition metals and their ions; (3) the application of crystal field theory in interrelating molecular geometry, magnetic properties, and color for coordination compounds; (4) combining coordination chemistry concepts with previous knowledge, such as acid-base chemistry and equilibrium.
The TMC Core Goal Skill:
Given a pair of complexes, be able to predict each of their magnetism (diamagnetic or paramagnetic), high spin/low spin, which will be more blue-shifted or red-shifted in UV-VIS Spectroscopy, and all possible isomers of the structures.
| Lectures | Readings | Assessments |
|---|---|---|
|
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Core Goals APP: Applications in Modern Chemistry
Unstable Nuclei: To gain a fundamental understanding of the composition of atoms with an emphasis on the atom’s nucleus. • To develop chemical literacy regarding isotopes, nuclear stability, radioactive decay, fission, fusion, and applications involving nuclear transformation.
The APP Core Goal Skill:
Given a complex system (polymer, materials, or biochemical), be able to make a list of concepts that need to be considered in order to fully understand/describe the system by analysis via MAT/SOL/AB/ECHEM/TMC Core Goals. These Assessments will be open note and open internet.
| Lectures | Readings | Assessments |
|---|---|---|
|
|
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Lecture 1: 5/17/2023
Introduction to each other and general goals of the course were discussed. The full walk-through General Chemistry prior knowledge was done focusing on energy, scale, and the way matter and electromagnetic radiation interact with each other. The conceptual focus for this lecture was on: Scale Matters. Think about waves of light (electromagnetic radiation), waves of matter (electronic orbitals), and the Coulombic interaction of these as the basis for understanding interactions. Moving charges have electromagnetic properties. The nucleus has charged particles (protons) so changing the number of protons greatly changes the reactivity and properites of an atom. Electrons have a charge and a very much lighter mass, so they have larger space to occupy with their wavelengths (based on deBroglie's wavelength equation). A review of the components of the atom was highlighted and the chart of Nuclides to discuss stable vs unstable nuclei. Nuclear reactions were discussed and how to balance them. Finally, a discussion of Einstein's famous equation \[E=mc^2\] was shown to be the source of immense amounts of energy from the Mass Deficit when nucleons combine to form atomic nuclei.
| Readings | ADAPT Homework |
|---|---|
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Final Exam: 6/28/2023 @ 2:00 p.m.-5:00 p.m in the DISCUSSION room
- TBD


10.12: Chi Square (Exercises)
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- Last updated
- Save as PDF
General Questions
Q1
Which of the two Chi Square distributions shown below (\(A\) or \(B\)) has the larger degrees of freedom? How do you know? (relevant section)
Q2
Twelve subjects were each given two flavors of ice cream to taste and then were asked whether they liked them. Two of the subjects liked the first flavor and nine of them liked the second flavor. Is it valid to use the Chi Square test to determine whether this difference in proportions is significant?  It is rolled \(25\) times with the following result:
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Outcome
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Frequency
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1
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9
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2
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4
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3
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1
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4
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8
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5
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3
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6
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0
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Conduct a significance test to see if the die is biased.
- What Chi Square value do you get and how many degrees of freedom does it have?
- What is the \(p\) value? (relevant section)
Q4
A recent experiment investigated the relationship between smoking and urinary incontinence. Of the \(322\) subjects in the study who were incontinent, \(113\) were smokers, \(51\) were former smokers, and \(158\) had never smoked. Of the \(284\) control subjects who were not incontinent, \(68\) were smokers, \(23\) were former smokers, and \(193\) had never smoked.
- Create a table displaying this data.
- What is the expected frequency in each cell?
- Conduct a significance test to see if there is a relationship between smoking and incontinence. What Chi Square value do you get? What \(p\) value do you get?
- What do you conclude? (relevant section)
Q5
At a school pep rally, a group of sophomore students organized a free raffle for prizes. They claim that they put the names of all of the students in the school in the basket and that they randomly drew \(36\) names out of this basket. Of the prize winners, \(6\) were freshmen, \(14\) were sophomores, \(9\) were juniors, and \(7\) were seniors. The results do not seem that random to you. You think it is a little fishy that sophomores organized the raffle and also won the most prizes. Your school is composed of \(30\%\) freshmen, \(25\%\) sophomores, \(25\%\) juniors, and \(20\%\) seniors.
- What are the expected frequencies of winners from each class?
- Conduct a significance test to determine whether the winners of the prizes were distributed throughout the classes as would be expected based on the percentage of students in each group. Report your Chi Square and \(p\) values.
- What do you conclude? (relevant section)
Q6
Some parents of the West Bay little leaguers think that they are noticing a pattern. There seems to be a relationship between the number on the kids' jerseys and their position. These parents decide to record what they see. The hypothetical data appear below. Conduct a Chi Square test to determine if the parents' suspicion that there is a relationship between jersey number and position is right. Report your Chi Square and p values. (relevant section)
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Infield
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Outfield
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Pitcher
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Total
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0-9
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12
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5
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5
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22
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10-19
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5
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10
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2
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17
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20+
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4
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4
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7
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15
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Total
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21
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19
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14
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54
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Q7
True/false: A Chi Square distribution with \(2\; df\) has a larger mean than a Chi Square distribution with \(12\; df\). (relevant section)
Q8
True/false: A Chi Square test is often used to determine if there is a significant relationship between two continuous variables. (relevant section)
Q9
True/false: Imagine that you want to determine if the spinner shown below is biased. You spin it \(50\) times and write down how many times the arrow lands in each section. You will reject the null hypothesis at the \(0.05\) level and determine that this spinner is biased if you calculate a Chi Square value of \(7.82\) or higher. (relevant section)
Questions from Case Studies
The following question uses data from the SAT and GPA (SG) case study.
Q10
Answer these items to determine if the math SAT scores are normally distributed. You may want to first standardize the scores. (relevant section)
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If these data were normally distributed, how many scores would you expect there to be in each of these brackets:
- smaller than \(1\) \(SD\) below the mean
- in between the mean and \(1\) \(SD\) below the mean
- in between the mean and \(1\) \(SD\) above the mean
- greater than \(1\) \(SD\) above the mean?
- How many scores are actually in each of these brackets?
- Conduct a Chi Square test to determine if the math SAT scores are normally distributed based on these expected and observed frequencies. (relevant section)
The following questions are from the Diet and Health (DH) case study.
Q11
(DH#3) Conduct a Pearson Chi Square test to determine if there is any relationship between diet and outcome. Report the Chi Square and \(p\) values and state your conclusions. (relevant section)
The following questions are from ARTIST (reproduced with permission).
Q12
A study compared members of a medical clinic who filed complaints with a random sample of members who did not complain. The study divided the complainers into two subgroups: those who filed complaints about medical treatment and those who filed nonmedical complaints. Here are the data on the total number in each group and the number who voluntarily left the medical clinic. Set up a two-way table. Analyze these data to see if there is a relationship between complaint (no, yes - medical, yes - nonmedical) and leaving the clinic (yes or no).
| No Complaint | Medical Complaint | Non Medical Complaint | |
| Total | 743 | 199 | 440 |
| Left | 22 | 26 | 28 |
Q13
Imagine that you believe there is a relationship between a person's eye color and where he or she prefers to sit in a large lecture hall. You decide to collect data from a random sample of individuals and conduct a chi-square test of independence. What would your two-way table look like? Use the information to construct such a table, and be sure to label the different levels of each category.
Q14
A geologist collects hand-specimen sized pieces of limestone from a particular area. A qualitative assessment of both texture and color is made with the following results. Is there evidence of association between color and texture for these limestones? Explain your answer.
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Color |
|||
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Texture |
Light |
Medium |
Dark |
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Fine |
4 |
20 |
8 |
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Medium |
5 |
23 |
12 |
|
Coarse |
21 |
23 |
4 |
Q15
Suppose that college students are asked to identify their preferences in political affiliation (Democrat, Republican, or Independent) and in ice cream (chocolate, vanilla, or strawberry). Suppose that their responses are represented in the following two-way table (with some of the totals left for you to calculate).
| Chocolate | Vanilla | Strawberry | Total | |
| Democrat | 26 | 43 | 13 | 82 |
| Republican | 45 | 12 | 8 | 65 |
| Independent | 9 | 13 | 4 | |
| Total | 68 | 25 | 173 |
- What proportion of the respondents prefer chocolate ice cream?
- What proportion of the respondents are Independents?
- What proportion of Independents prefer chocolate ice cream?
- What proportion of those who prefer chocolate ice cream are Independents?
- Analyze the data to determine if there is a relationship between political party preference and ice cream preference.
Q16
NCAA collected data on graduation rates of athletes in Division I in the mid \(1980s\). Among \(2,332\) men, \(1,343\) had not graduated from college, and among \(959\) women, \(441\) had not graduated.
- Set up a two-way table to examine the relationship between gender and graduation.
- Identify a test procedure that would be appropriate for analyzing the relationship between gender and graduation. Carry out the procedure and state your conclusion.
Select Answers
S3
- \(\text{Chi Square} = 16.0\), \(df = 5\)
S4
- Incontinent/Smoker cell: \(96.2\)
S5
- \(p = 0.18\)
S6
\(\text{Chi Square} = 10.2\)
S10
-
- Scores smaller than \(1\) \(SD\) below the mean: \(24\)
S11
\(\text{Chi Square} = 16.6\)


27.E: Introduction to Animal Diversity (Exercises)
27.1: Features of the Animal Kingdom
Review Questions
Which of the following is not a feature common to most animals?
- development into a fixed body plan
- asexual reproduction
- specialized tissues
- heterotrophic nutrient sourcing
- Answer
-
B
During embryonic development, unique cell layers develop and distinguish during a stage called ________.
- the blastula stage
- the germ layer stage
- the gastrula stage
- the organogenesis stage
- Answer
-
C
Which of the following phenotypes would most likely be the result of a Hox gene mutation?
- abnormal body length or height
- two different eye colors
- the contraction of a genetic illness
- two fewer appendages than normal
- Answer
-
D
Free Response
Why might the evolution of specialized tissues be important for animal function and complexity?
- Answer
-
The development of specialized tissues affords more complex animal anatomy and physiology because differentiated tissue types can perform unique functions and work together in tandem to allow the animal to perform more functions. For example, specialized muscle tissue allows directed and efficient movement, and specialized nervous tissue allows for multiple sensory modalities as well as the ability to respond to various sensory information; these functions are not necessarily available to other non-animal organisms.
Describe and give examples of how humans display all of the features common to the animal kingdom.
- Answer
-
Humans are multicellular organisms. They also contain differentiated tissues, such as epithelial, muscle, and nervous tissue, as well as specialized organs and organ systems. As heterotrophs, humans cannot produce their own nutrients and must obtain them by ingesting other organisms, such as plants, fungi, and animals. Humans undergo sexual reproduction, as well as the same embryonic developmental stages as other animals, which eventually lead to a fixed and motile body plan controlled in large part by Hox genes.
How have Hox genes contributed to the diversity of animal body plans?
- Answer
-
Altered expression of homeotic genes can lead to major changes in the morphology of the individual. Hox genes can affect the spatial arrangements of organs and body parts. If a Hox gene was mutated or duplicated, it could affect where a leg might be on a fruit fly or how far apart a person’s fingers are.
27.2: Features Used to Classify Animals
Review Questions
Which of the following organism is most likely to be a diploblast?
- sea star
- shrimp
- jellyfish
- insect
- Answer
-
C
Which of the following is not possible?
- radially symmetrical diploblast
- diploblastic eucoelomate
- protostomic coelomate
- bilaterally symmetrical deuterostome
- Answer
-
B
An animal whose development is marked by radial cleavage and enterocoely is ________.
- a deuterostome
- an annelid or mollusk
- either an acoelomate or eucoelomate
- none of the above
- Answer
-
A
Free Response
Using the following terms, explain what classifications and groups humans fall into, from the most general to the most specific: symmetry, germ layers, coelom, cleavage, embryological development.
- Answer
-
Humans have body plans that are bilaterally symmetrical and are characterized by the development of three germ layers, making them triploblasts. Humans have true coeloms and are thus eucoelomates. As deuterostomes, humans are characterized by radial and indeterminate cleavage.
Explain some of the advantages brought about through the evolution of bilateral symmetry and coelom formation.
- Answer
-
The evolution of bilateral symmetry led to designated head and tail body regions, and promoted more efficient mobility for animals. This improved mobility allowed for more skillful seeking of resources and prey escaping from predators. The appearance of the coelom in coelomates provides many internal organs with shock absorption, making them less prone to physical damage from bodily assault. A coelom also gives the body greater flexibility, which promotes more efficient movement. The relatively loose placement of organs within the coelom allows them to develop and grow with some spatial freedom, which promoted the evolution of optimal organ arrangement. The coelom also provides space for a circulatory system, which is an advantageous way to distribute body fluids and gases.
27.3: Animal Phylogeny
Review Questions
Consulting the modern phylogenetic tree of animals, which of the following would not constitute a clade?
- deuterostomes
- lophotrochozoans
- Parazoa
- Bilateria
- Answer
-
C
Which of the following is thought to be the most closely related to the common animal ancestor?
- fungal cells
- protist cells
- plant cells
- bacterial cells
- Answer
-
B
As with the emergence of the Acoelomorpha phylum, it is common for ____ data to misplace animals in close relation to other species, whereas ____ data often reveals a different and more accurate evolutionary relationship.
- molecular : morphological
- molecular : fossil record
- fossil record : morphological
- morphological : molecular
- Answer
-
D
Free Response
Describe at least two major changes to the animal phylogenetic tree that have come about due to molecular or genetic findings.
- Answer
-
Two new clades that comprise the two major groups of protostomes are called the lophotrochozoans and the ecdysozoans. The formation of these two clades came about through molecular research from DNA and protein data. Also, the novel phylum of worm called Acoelomorpha was determined due to molecular data that distinguished them from other flatworms.
How is it that morphological data alone might lead scientists to group animals into erroneous evolutionary relationships?
- Answer
-
In many cases, morphological similarities between animals may be only superficial similarities and may not indicate a true evolutionary relationship. One of the reasons for this is that certain morphological traits can evolve along very different evolutionary branches of animals for similar ecological reasons.
27.4: The Evolutionary History of the Animal Kingdom
Review Questions
Which of the following periods is the earliest during which animals may have appeared?
- Ordovician period
- Cambrian period
- Ediacaran period
- Cryogenian period
- Answer
-
D
What type of data is primarily used to determine the existence and appearance of early animal species?
- molecular data
- fossil data
- morphological data
- embryological development data
- Answer
-
B
The time between 542–488 million years ago marks which period?
- Cambrian period
- Silurian period
- Ediacaran period
- Devonian period
- Answer
-
A
Until recent discoveries suggested otherwise, animals existing before the Cambrian period were believed to be:
- small and ocean-dwelling
- small and non-motile
- small and soft-bodied
- small and radially symmetrical or asymmetrical
- Answer
-
C
Plant life first appeared on land during which of the following periods?
- Cambrian period
- Ordovician period
- Silurian period
- Devonian period
- Answer
-
B
Approximately how many mass extinction events occurred throughout the evolutionary history of animals?
- 3
- 4
- 5
- more than 5
- Answer
-
D
Free Response
Briefly describe at least two theories that attempt to explain the cause of the Cambrian explosion.
- Answer
-
One theory states that environmental factors led to the Cambrian explosion. For example, the rise in atmospheric oxygen and oceanic calcium levels helped to provide the right environmental conditions to allow such a rapid evolution of new animal phyla. Another theory states that ecological factors such as competitive pressures and predator-prey relationships reached a threshold that supported the rapid animal evolution that took place during the Cambrian period.
How is it that most, if not all, of the extant animal phyla today evolved during the Cambrian period if so many massive extinction events have taken place since then?
- Answer
-
It is true that multiple mass extinction events have taken place since the Cambrian period, when most currently existing animal phyla appeared, and the majority of animal species were commonly wiped out during these events. However, a small number of animal species representing each phylum were usually able to survive each extinction event, allowing the phylum to continue to evolve rather than become altogether extinct.


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-
active immunity
-
immunity that results from the production of antibodies to help protect the body from a pathogen that has entered the body
-
airborne precaution
-
a practice indicated for patients known or suspected to be infected with pathogens that are transmitted through droplet nuclei
-
airborne transmission
-
the spread of pathogens that are carried by dust or the nuclei of an evaporated droplet and remain suspended in the air
-
antibody (also, immunoglobulin)
-
a protein created in the body in response to an antigen in order to fight the identified substance or toxin
-
antigen
-
anything the immune system recognizes as a foreign object or substance and subsequently initiates formation of antibodies
-
asepsis
-
the state of being free from disease-causing contaminants
-
chain of infection
-
the six stages of infection that must be present for the transmission of a pathogen from one place or person to another
-
contact precaution
-
a practice implemented for patients who have known or suspected infections that can be transmitted through direct skin-to-skin contact or indirect contact with environmental surfaces or equipment
-
convalescent period
-
the time when recovery and healing begin
-
direct transmission
-
the spread of a pathogen that is transmitted directly from an infected person
-
disinfection
-
the process of removing microorganisms and disease-causing pathogens from inanimate objects
-
droplet precaution
-
a practice implemented when patients are known or suspected to be infected with pathogens transmitted by respiratory droplets from coughing, sneezing, and talking
-
droplet transmission
-
the spread of a pathogen that travels through a spray of water droplets that are released when an infected person coughs, sneezes, or talks
-
fever
-
a rise in core body temperature
-
healthcare-associated infection (HAI)
-
an infection that is acquired while receiving medical treatment in a healthcare facility
-
illness period
-
the third stage of infection, which starts when a person experiences the specific signs and symptoms of a disease
-
incubation period
-
the initial stage of infection when a pathogen successfully enters into a new host
-
indirect transmission
-
the spread of a pathogen to a new host through an intermediary, such as the air, food, water, animals, or objects
-
infection control
-
the discipline of stopping or preventing the spread of infectious agents
-
inflammatory response
-
one of the first responses when a pathogen breaches the nonspecific innate immune system; it aids in the recruitment of cellular defenses to remove pathogens
-
medical asepsis
-
techniques and procedures used to decrease the potential for the spread of microorganisms and infection
-
medical isolation gown
-
a garment worn in the healthcare setting as a broad barrier against blood or bodily fluids
-
microorganism
-
a single-celled organism that is microscopic in size and too small to be seen by the naked eye
-
multidrug-resistant organisms
-
a bacterium that has become resistant to certain antibiotics
-
N95 respirator mask
-
a tight-fitting mask that protects the wearer from very small particles that float in the air
-
normal flora
-
microorganisms that can live on another living organism or inanimate object without causing diseases or complications
-
passive immunity
-
the result of antibodies being passed from one person to another
-
pathogen
-
any type of microorganism that causes disease to its host
-
personal protective equipment (PPE)
-
a means of barrier protection for eyes, nose, mouth, and skin from exposure to blood and other potentially infectious bodily fluids and materials
-
portal of entry
-
the site at which a pathogen enters the susceptible host
-
portal of exit
-
the path by which the pathogen leaves the reservoir
-
prodromal period
-
the second stage of infection, which begins at the initial appearance of mild or vague symptoms
-
reservoir
-
the habitat or source of the pathogen that provides a place for survival and growth
-
sanitization
-
the removal of visible soil from objects and materials
-
sepsis
-
the body’s extreme response to a widespread infection
-
sharps
-
needles and other objects, such as lancets, razor blades, scissors, clamps, pins, staples, and glass items
-
standard precaution
-
the first tier of precautions to protect healthcare staff and patients
-
sterile gloves
-
a glove that is free from microorganisms and individually packaged as a pair to remain free of pathogens
-
sterile technique
-
practices and procedures that inhibit microorganisms in a specific environment and prevent contamination
-
sterilization
-
a procedure that eliminates all microorganisms in or on an object
-
surgical asepsis
-
the absence of all microorganisms within any type of invasive procedure; requires the use of sterile technique
-
surgical gown
-
a garment worn by healthcare personnel during a surgical procedure to protect both the patient and the personnel from transmission of pathogens
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surgical mask
-
a mask that prevents transmission of large particle droplet matter generated through coughing or splash-generating procedures
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susceptible host
-
the organism that is susceptible to or at risk for infection
-
transmission
-
the method that a pathogen uses to spread from one host to another
-
transmission-based precaution
-
a practice implemented in situations where standard precautions are not adequate to prevent infection transmission
-
vector transmission
-
the spread of a pathogen that occurs when blood-feeding arthropods infect animals or humans
-
zoonosis
-
an infectious disease that can naturally transmit from animal to human


4.4: Length of a Vector
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- Last updated
- Save as PDF
Learning Objectives
- Find the length of a vector and the distance between two points in \(\mathbb{R}^n\).
- Find the corresponding unit vector to a vector in \(\mathbb{R}^n\).
In this section, we explore what is meant by the length of a vector in \(\mathbb{R}^n\). We develop this concept by first looking at the distance between two points in \(\mathbb{R}^n\).
First, we will consider the concept of distance for \(\mathbb{R}\), that is, for points in \(\mathbb{R}^1\). Here, the distance between two points \(P\) and \(Q\) is given by the absolute value of their difference. We denote the distance between \(P\) and \(Q\) by \(d(P,Q)\) which is defined as \[d(P,Q) = \sqrt{ \left( P-Q\right) ^{2}} \label{distance1}\]
Consider now the case for \(n=2\), demonstrated by the following picture.
There are two points \(P =\left( p_{1},p_{2}\right)\) and \(Q = \left(q_{1},q_{2}\right)\) in the plane. The distance between these points is shown in the picture as a solid line. Notice that this line is the hypotenuse of a right triangle which is half of the rectangle shown in dotted lines. We want to find the length of this hypotenuse which will give the distance between the two points. Note the lengths of the sides of this triangle are \(\left| p_{1}-q_{1}\right|\) and \(\left| p_{2}-q_{2}\right|\), the absolute value of the difference in these values. Therefore, the Pythagorean Theorem implies the length of the hypotenuse (and thus the distance between \(P\) and \(Q\)) equals \[\left( \left| p_{1}-q_{1}\right| ^{2}+\left| p_{2}-q_{2}\right| ^{2}\right) ^{1/2}=\left( \left( p_{1}-q_{1}\right) ^{2}+\left( p_{2}-q_{2}\right) ^{2}\right) ^{1/2} \label{distance2}\]
Now suppose \(n=3\) and let \(P = \left( p_{1},p_{2},p_{3}\right)\) and \(Q = \left( q_{1},q_{2},q_{3}\right)\) be two points in \(\mathbb{R}^{3}.\) Consider the following picture in which the solid line joins the two points and a dotted line joins the points \(\left( q_{1},q_{2},q_{3}\right)\) and \(\left( p_{1},p_{2},q_{3}\right) .\)
Here, we need to use Pythagorean Theorem twice in order to find the length of the solid line. First, by the Pythagorean Theorem, the length of the dotted line joining \(\left( q_{1},q_{2},q_{3}\right)\) and \(\left( p_{1},p_{2},q_{3}\right)\) equals \[\left( \left( p_{1}-q_{1}\right) ^{2}+\left( p_{2}-q_{2}\right) ^{2}\right) ^{1/2}\nonumber \] while the length of the line joining \(\left( p_{1},p_{2},q_{3}\right)\) to \(\left( p_{1},p_{2},p_{3}\right)\) is just \(\left| p_{3}-q_{3}\right| .\) Therefore, by the Pythagorean Theorem again, the length of the line joining the points \(P = \left( p_{1},p_{2},p_{3}\right)\) and \(Q = \left( q_{1},q_{2},q_{3}\right)\) equals \[\left( \left( \left( \left( p_{1}-q_{1}\right) ^{2}+\left( p_{2}-q_{2}\right) ^{2}\right) ^{1/2}\right) ^{2}+\left( p_{3}-q_{3}\right) ^{2}\right) ^{1/2}\nonumber \] \[=\left( \left( p_{1}-q_{1}\right) ^{2}+\left( p_{2}-q_{2}\right) ^{2}+\left( p_{3}-q_{3}\right) ^{2}\right) ^{1/2} \label{distance3}\]
This discussion motivates the following definition for the distance between points in \(\mathbb{R}^n\).
Definition \(\PageIndex{1}\): Distance Between Points
Let \(P=\left( p_{1},\cdots ,p_{n}\right)\) and \(Q=\left( q_{1},\cdots ,q_{n}\right)\) be two points in \(\mathbb{R}^{n}\). Then the distance between these points is defined as \[\text{ distance between }P\text{ and } Q\text{ } = d( P, Q ) = \left( \sum_{k=1}^{n}\left\vert p_{k}-q_{k}\right\vert ^{2}\right) ^{1/2}\nonumber \] This is called the distance formula . We may also write \(\left\vert P - Q \right\vert\) as the distance between \(P\) and \(Q\).
From the above discussion, you can see that Definition \(\PageIndex{1}\) holds for the special cases \(n=1,2,3\), as in Equations \(\eqref{distance1}\), \(\eqref{distance2}\), \(\eqref{distance3}\). In the following example, we use Definition \(\PageIndex{1}\) to find the distance between two points in \(\mathbb{R}^4\).
Example \(\PageIndex{1}\): Distance Between Points
Find the distance between the points \(P\) and \(Q\) in \(\mathbb{R}^{4}\), where \(P\) and \(Q\) are given by \[P=\left( 1,2,-4,6\right)\nonumber \] and \[Q=\left( 2,3,-1,0\right)\nonumber \]
Solution
We will use the formula given in Definition \(\PageIndex{1}\) to find the distance between \(P\) and \(Q\). Use the distance formula and write \[d(P,Q)= \left( \left( 1-2\right) ^{2}+\left( 2-3\right) ^{2}+\left( -4-\left( -1\right) \right) ^{2}+\left( 6-0\right)^{2}\right) ^{\frac{1}{2}} = 47\nonumber \]
Therefore, \(d( P,Q) = \sqrt{47}.\)
There are certain properties of the distance between points which are important in our study. These are outlined in the following theorem.
Theorem \(\PageIndex{1}\): Properties of Distance
Let \(P\) and \(Q\) be points in \(\mathbb{R}^n\), and let the distance between them, \(d( P, Q)\), be given as in Definition \(\PageIndex{1}\) . Then, the following properties hold.
- \(d( P, Q) = d( Q, P)\)
- \(d( P, Q) \geq 0\), and equals 0 exactly when \(P = Q.\)
There are many applications of the concept of distance. For instance, given two points, we can ask what collection of points are all the same distance between the given points. This is explored in the following example.
Example \(\PageIndex{2}\): The Plane Between Two Points
Describe the points in \(\mathbb{R}^3\) which are at the same distance between \(\left( 1,2,3\right)\) and \(\left( 0,1,2\right) .\)
Solution
Let \(P = \left( p_1 , p_2, p_3\right)\) be such a point. Therefore, \(P\) is the same distance from \(\left( 1,2,3\right)\) and \(\left( 0,1,2\right) .\) Then by Definition \(\PageIndex{1}\) , \[\sqrt{\left( p_1 -1\right) ^{2}+\left( p_2 -2\right) ^{2}+\left( p_3-3\right) ^{2}}= \sqrt{\left( p_1 - 0 \right)^{2}+\left( p_2-1\right) ^{2}+\left( p_3-2\right) ^{2}}\nonumber \] Squaring both sides we obtain \[\left( p_1 -1\right) ^{2}+\left( p_2 -2\right) ^{2}+\left( p_3 -3\right) ^{2}=p_1^{2}+\left( p_2-1\right) ^{2}+\left( p_3 -2\right) ^{2}\nonumber \] and so \[ \ p_1^{2}-2p_1+14+p_2^{2}-4p_2+p_3^{2}-6p_3=p_1^{2}+p_2^{2}-2p_2+5+p_3^{2}-4p_3\nonumber \] Simplifying, this becomes \[-2p_1+14-4p_2-6p_3=-2p_2+5-4p_3\nonumber \] which can be written as \[2p_1+2p_2+2p_3=-9 \label{distanceplane}\] Therefore, the points \(P = \left( p_1,p_2,p_3\right)\) which are the same distance from each of the given points form a plane whose equation is given by \(\eqref{distanceplane}\).
We can now use our understanding of the distance between two points to define what is meant by the length of a vector. Consider the following definition.
Definition \(\PageIndex{2}\): Length of a Vector
Let \(\vec{u} = \left[ u_{1} \cdots u_{n} \right]^T\) be a vector in \(\mathbb{R}^n\). Then, the length of \(\vec{u}\), written \(\| \vec{u} \|\) is given by \[\| \vec{u} \| = \sqrt{ u_{1}^2 + \cdots + u_{n}^2}\nonumber \]
This definition corresponds to Definition \(\PageIndex{1}\) , if you consider the vector \(\vec{u}\) to have its tail at the point \(0 = \left( 0, \cdots ,0 \right)\) and its tip at the point \(U = \left(u_1, \cdots, u_n \right)\). Then the length of \(\vec{u}\) is equal to the distance between \(0\) and \(U\), \(d(0,U)\). In general, \(d(P,Q)=||\vec{PQ}||\).
Consider Example \(\PageIndex{1}\) . By Definition \(\PageIndex{2}\) , we could also find the distance between \(P\) and \(Q\) as the length of the vector connecting them. Hence, if we were to draw a vector \(\overrightarrow{PQ}\) with its tail at \(P\) and its point at \(Q\), this vector would have length equal to \(\sqrt{47}\).
We conclude this section with a new definition for the special case of vectors of length \(1\).
Definition \(\PageIndex{3}\): Unit Vector
Let \(\vec{u}\) be a vector in \(\mathbb{R}^{n}\). Then, we call \(\vec{u}\) a unit vector if it has length 1, that is if \[\| \vec{u} \| = 1\nonumber \]
Let \(\vec{v}\) be a vector in \(\mathbb{R}^{n}\). Then, the vector \(\vec{u}\) which has the same direction as \(\vec{v}\) but length equal to \(1\) is the corresponding unit vector of \(\vec{v}\). This vector is given by \[\vec{u} = \frac{1}{\| \vec{v} \|} \vec{v}\nonumber \]
We often use the term normalize to refer to this process. When we normalize a vector, we find the corresponding unit vector of length \(1\). Consider the following example.
Example \(\PageIndex{3}\): Finding a Unit Vector
Let \(\vec{v}\) be given by \[\vec{v} = \left[ \begin{array}{rrr} 1 & -3 & 4 \end{array} \right]^T\nonumber \] Find the unit vector \(\vec{u}\) which has the same direction as \(\vec{v}\).
Solution
We will use Definition \(\PageIndex{3}\) to solve this. Therefore, we need to find the length of \(\vec{v}\) which, by Definition \(\PageIndex{2}\) is given by \[\| \vec{v} \| = \sqrt{ v_{1}^2 + v_{2}^2+ v_{3}^2}\nonumber \] Using the corresponding values we find that \[\begin{aligned} \| \vec{v} \| &= \sqrt{ 1^2 + \left(-3 \right)^2 + 4^2} \\ &= \sqrt{ 1 + 9 + 16} \\ &= \sqrt{26} \end{aligned}\] In order to find \(\vec{u}\), we divide \(\vec{v}\) by \(\sqrt{26}\). The result is \[\begin{aligned} \vec{u} &= \frac{1}{\| \vec{v} \|} \vec{v} \\ &= \frac{1}{\sqrt{26}} \left[ \begin{array}{rrr} 1 & -3 & 4 \end{array} \right]^T \\ &= \left[ \begin{array}{rrr} \frac{1}{\sqrt{26}} & -\frac{3}{\sqrt{26}} & \frac{4}{\sqrt{26}} \end{array} \right]^T\end{aligned}\]
You can verify using the Definition \(\PageIndex{1}\) that \(\| \vec{u} \| = 1\).


7.1: Thumbs Up or Down?
By the end of this section, you will be able to:
- Explain the role of the review genre in personal, professional, and academic contexts.
- Articulate what differentiates the review genre from other genres.
Developing evaluation skills can help you in everyday life. Just about anything you buy or use will require you to evaluate a range of choices based on criteria that are important to you. For example, writing a good paper or making a good presentation necessitates locating and evaluating sources. See Annotated Bibliography: Gathering, Evaluating, and Documenting Sources for more information. You also may be asked to evaluate the effectiveness of your courses at the end of the semester. Or you may be asked to evaluate the work of your peers to help them revise their compositions. In the professional world, you may be asked to evaluate solutions to problems, employees you supervise, and in some cases, even yourself. Evaluating effectively makes you not only a better consumer but also a better student, employee, and possible supervisor.
Using Evidence to Make a Judgment
When you review or evaluate something, the end result is your judgment about it. Should your readers see the film? Are the food and service good at the restaurant? Should you use this source in your essay? Does your employee deserve a raise? Making a clear judgment about the subject of your evaluation provides guidance for the actions that audience members may take on the basis of the information you provide.
Ultimately, your judgment is your opinion. For example, it is expected that some people will love Avengers: Endgame (2019) and others will not. In fact, because some people may disagree with you, reviews provide a perfect opportunity to use evidence to defend your judgment. You are probably familiar with some ways in which reviewers present their judgments about their subjects. Reviews on Facebook , Google , and Yelp have a star rating system (the more stars the better). The film review site Rotten Tomatoes shows the percentage of reviewers that recommend the film. The review site The AV Club rates films and TV episodes by using an A-to-F grading scale.
While it is important to present your overall judgment in a review, a simple “I liked it” or “I didn’t like it” is not enough to help your audience make their own judgments. It is also important to explain why you arrived at the judgment you did. Think about some of the titles of reviews you have seen online. One might simply read “DIRTY!” about an experience staying in a hotel. Other reviews might present a thesis , or debatable main idea, as a title, such as Slate culture critic Willa Paskin’s “In Its Immensely Satisfying Season Finale, Game of Thrones Became the Show It Had Always Tried Not to Be.” In both examples, the title provides an overall reason for the author’s judgment.
Although a simple rating might be effective when reviewing a business, reviews of creative works such as films, TV shows, visual arts, and books are more complex. Critics —professional writers who review creative works—like Willa Paskin try to review their subjects and at the same time analyze their subjects’ cultural significance. In addition to providing an overall judgment, critics guide audiences on how to view and understand a work within a larger cultural context. Critics provide this guidance by answering questions such as these:
- In what genre would I place this work? Why?
- What has this work contributed to its genre that other works have not?
- How does the creator (or creators) of this work show they understand the culture (audience) that will view the work?
- How does this work reflect the time in which it was created?
People look to critics not only to judge the overall quality of a work but also to gain insights about it.
Other writing genres feature characteristics similar to the review genre, using criteria and evidence to analyze and sometimes evaluate the effectiveness of written or visual text: Rhetorical Analysis: Interpreting the Art of Rhetoric, Print or Textual Analysis: What You Read, and Image Analysis: Writing About What You See.


11.0: Prelude to Fluid Statics
Much of what we value in life is fluid: a breath of fresh winter air; the hot blue flame in our gas cooker; the water we drink, swim in, and bathe in; the blood in our veins. What exactly is a fluid? Can we understand fluids with the laws already presented, or will new laws emerge from their study? The physical characteristics of static or stationary fluids and some of the laws that govern their behavior are the topics of this chapter. Fluid Dynamics and Its Biological and Medical Applications explores aspects of fluid flow.
Contributors and Attributions
Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a
Creative Commons Attribution License (by 4.0)
.


29.5: Chapter Summary
Occupational and environmental health nursing focuses on providing preventive health care, health promotion, and health restoration in a safe and healthy work environment. OHNs require specialized competencies depending on the industry; however, the scope of practice has common overarching concepts of community health, law, economy, politics, policy, and regulatory issues. OSHA creates and enforces workplace safety standards to ensure safe and healthy conditions for workers. NIOSH is the research agency tasked with studying worker safety and health. NIOSH provides evidence for many of the standards that OSHA enforces. OHNs’ roles include clinician, educator, case manager, corporate director, consultant, and researcher with diverse work settings including hospitals, academia, government, military, and industrial factories.
School nursing protects and promotes student health by facilitating optimal development and academic success. School nurses are health care and education leaders who advocate for student-centered care, coordinate care, and collaborate with school administrators, staff, teachers, families, and the interprofessional health care team. They are change agents, improving the health of children, families, and communities. Roles include clinical care provider, health educator, and advocate. Two established frameworks, the Framework for 21st Century School Nursing Practice and the Whole School, Whole Community, Whole Child Model , guide school nursing practice. Common health concerns the school nurse encounters are asthma, diabetes, epilepsy, food allergies and anaphylaxis, obesity, oral health issues, behavioral problems, learning difficulties, teen pregnancy, STIs, and substance misuse.
Correctional nursing provides care to clients in the criminal justice system, jails, prisons, juvenile detention centers, and substance-misuse treatment centers. Correctional nurses must defer to correctional officers and safety protocols. They use a broad range of nursing skills to care for individuals across the justice system, including incarceration, probation, parole, treatment programs, individuals housed in immigration and customs facilities, and those transitioning between settings. They perform many roles in various settings, such as restricted housing, primary care clinics, urgent care clinics, inpatient infirmaries, community-based facilities, and palliative and hospice care settings.
Public health nursing promotes and protects a population’s health by synthesizing knowledge from nursing, social, and public health sciences and applying it for social betterment. The government sector of public health includes public health agencies at the federal and state levels, including the District of Columbia, with almost 3,000 local governments and over 500 federally recognized tribal agencies. SHDs’ responsibilities include policy development, resource stewardship, legal compliance, partner engagement, continuous improvement, and oversight. SHDs provide population-based public health services focusing on primary, secondary, and tertiary prevention. LHDs’ roles and scope of authority are contingent on state policy and their governing relationship with the state. LHDs coordinate public health activities in their jurisdiction and follow the SHDs’ rules and regulations.


24.1: The Muromachi Period
Zen Ink Painting
During the Muromachi period (1333–1578), Zen Buddhism played an influential role in the development of Zen ink painting in Japan.
Distinguish the techniques of the Yamato-e, Sumi-e, Sansuiga, and Shigajiku styles of Japanese Zen Ink painting
Key Points
- The development of the great Zen monasteries in Kamakura and Kyoto had a great influence on the visual arts of the Muromachi period.
- The foremost painter of the new Sumi-e style was Sesshū Tōyō (1420–1506), whose most dramatic works were completed in the Chinese splashed-ink (Haboku) style.
- The Sumi-e style was highly influenced by calligraphy, employing the same tools and style as well as its Zen philosophy.
- By the end of the 14th century, monochrome landscape paintings (sansuiga) had found patronage by the ruling Ashikaga family and were the preferred genre among Zen painters, gradually evolving from its Chinese roots to a more Japanese style.
- Another style that developed during the Muromachi period was Shigajiku (詩), or paintings accompanied by poetry; this style had its roots in China, where painting and poetry were seen as inherently connected.
Key Terms
- calligraphy : The art of writing letters and words with decorative strokes.
- Koan : A story, dialogue, question, or statement, which is used in Zen practice to provoke the “great doubt” and test a student’s progress.
- shogunate : A hereditary military dictator in Japan during the period from 1185 to 1868 (with exceptions).
Background: The Muromachi Period
During the Muromachi period (1333–1578), also known as the Ashikaga period, a profound change took place in Japanese culture. The Ashikaga clan took control of the shogunate and moved its headquarters back to Kyoto, to the Muromachi district of the city. With the return of government to the capital, the popularizing trends of the Kamakura period came to an end, and cultural expression took on a more aristocratic, elitist character. During the Muromachi Period, Zen Buddhism rose to prominence—especially among the elite Samurai class, who embraced the Zen values of personal discipline, concentration, and self-development.
Impact on the Arts
The establishment of the great Zen monasteries in Kamakura and Kyoto had a major impact on the visual arts. Because of secular ventures and trading missions to China organized by Zen temples, many Chinese paintings and objects of art were imported into Japan, profoundly influencing Japanese artists working for Zen temples and the shogunate. These imports not only changed the subject matter of painting, but they also modified the use of color; the bright colors of Yamato-e yielded to the monochromes of painting in the Chinese manner of Sui-boku-ga (水) or Sumi-e (墨). This style mainly used only black ink—the same as used in East Asian calligraphy.
Sesshū Tōyō and the Haboku Style
The foremost painter of the new Sumi-e style was Sesshū Tōyō (1420–1506), a Rinzai priest who traveled to China in 1468–69 and studied contemporary Ming painting. Some of his most dramatic works are in the Chinese splashed-ink (Haboku) style. Upon returning to Japan, Sesshū built himself a studio and established a large following; these painters are now referred to as the Unkoku-rin school or School of Sesshū.
To make one of the calligraphic and highly stylized Haboku paintings, the painter would visualize the image and then make swift broad strokes onto the paper, resulting in a splashed and abstract composition. This was all done with meditative concentration. This impressionistic style of painting was supposed to capture the true nature of the subject. The Sumi-e style was highly influenced by calligraphy, using the same tools and style as well as its Zen philosophy. To paint in this style, the practitioner had to clear his mind and apply the brushstrokes without too much thinking, termed mushin (無) by the Japanese philosopher Nishida Kitaro. The concept of mushin is central to many Japanese arts, including the art of the sword, archery, and the tea ceremony.
Sansuiga
By the end of the 14th century, monochrome landscape paintings (sansuiga) had found patronage by the ruling Ashikaga family and became the preferred genre among Zen painters, gradually evolving from their Chinese roots to a more Japanese style. An important landscape painter during this period was Tenshō Shūbun, a monk at the Kyoto temple of Shōkoku-ji who traveled to Korea and studied under Chinese painters. He returned to Japan in 1404 and settled in Kyoto, then the capital city. He became director of the court painting bureau that had been established by Ashikaga shoguns, who were influential art patrons. Shūbun’s best known landscape painting, designated as a National Treasure in Japan, is Reading in a Bamboo Grove , now kept in the Tokyo National Museum.
Shigajiku
Another style that developed in the Muromachi period is Shigajiku (詩). This is usually a painting accompanied by poetry and has its roots in China, where painting and poetry were seen as inherently connected. This style grew out of literary circles; an artist would usually be given a subject to paint, and the poets would compose accompanying verses to be written above the work.
A famous example is the scroll Catching a Catfish with a Gourd (Hyōnen-zu 瓢), located at Taizō-in, Myōshin-ji, Kyoto. Created by the priest-painter Josetsu (c. 1386–1428), it includes 31 verses of many Zen priests inscribed above the painting. In the foreground of the painting, a man is depicted on the bank of a stream holding a small gourd and looking at a large slithery catfish. Mist fills the middle ground, and the background mountains appear to be far in the distance. The painting was commissioned by the 4th Shogun of the Muromachi Period, Ashikaga Yoshimochi (1386–1428), and was based on the nonsensical riddle: “How do you catch a catfish with a gourd?” The painting and accompanying poems capture both the playfulness and the perplexing nature of Zen buddhist Koans, which were supposed to aid the Zen practitioner in their meditation.
In the late Muromachi period, ink painting had migrated out of the Zen monasteries into the art world in general. Artists from the Kano School and the Ami School adopted the style and themes but introduced a more plastic and decorative effect that would continue into modern times.
Zen Dry Rock Gardens
Zen dry rock gardens were created at temples of Zen Buddhism during the Muromachi Period to imitate the intimate essence of nature.
Describe the features of the Zen dry rock gardens of the Muromachi Period
Key Points
- The Muromachi Period in Japan was characterized by political rivalaries that frequently led to wars, but also by an extraordinary flourishing of Japanese culture. It saw the beginning of Noh theater, the Japanese tea ceremony, the shoin style of Japanese architecture, and the zen garden.
- In Kyoto in the 14th and 15th century, a new kind of garden designed to stimulate meditation began to appear at the important zen temples.
- These new zen dry rock gardens were usually relatively small, surrounded by a wall, and meant to be seen while seated from a single viewpoint outside the garden.
- The invention of the zen garden was closely connected with developments in Japanese ink landscape paintings. Japanese painters such as Sesshū Tōyō (1420–1506) and Soami (1472–1525) greatly simplified their views of nature, showing only the most essential aspects of nature.
- The most famous of all zen gardens in Kyoto is Ryōan-ji, built in the late 15th century when, for the first time, the zen garden became purely abstract.
Key Terms
- Noh theater : A major form of classical Japanese musical drama that has been performed since the 14th century; many characters are masked, with men playing both male and female roles.
- shoin : A type of audience hall in Japanese architecture that was developed during the Muromachi period; the term originally meant a study and a place for lectures on the sūtra within a temple, but later it came to mean just a drawing room or study.
Background: The Muromachi Period
The Muromachi Period in Japan, which took place at roughly the same time as the Renaissance in Europe, was characterized by political rivalries that frequently led to wars. However, it was also characterized by an extraordinary flourishing of Japanese culture. It saw the beginning of Noh theater, the Japanese tea ceremony, the shoin style of Japanese architecture, and the zen garden.
A New Kind of Garden
Zen Buddhism was introduced into Japan at the end of the 12th century. It quickly achieved a wide following, particularly among the Samurai class and war lords, who admired its doctrine of self-discipline. The gardens of the early zen temples in Japan resembled Chinese gardens at the time, with lakes and islands. However, in Kyoto in the 14th and 15th century, a new kind of garden appeared at the important zen temples. These zen gardens were designed to stimulate meditation. “Nature, if you made it expressive by reducing it to its abstract forms, could transmit the most profound thoughts by its simple presence,” Michel Baridon, a well-known researcher, wrote. “The compositions of stone, already common China, became in Japan, veritable petrified landscapes, which seemed suspended in time, as in a certain moment of Noh theater, which dates to the same period.”
Saihō-ji
The first garden to begin the transition to this new style is considered by many experts to be Saihō-ji, The Temple of the Perfumes of the West—popularly known as Koke-dera, the Moss Garden—in the western part of Kyoto. The Buddhist monk and zen master Musō Kokushi transformed a Buddhist temple into a zen monastery in 1334 and built the gardens. The lower garden of Saihō-ji is in the traditional Heian Period style: a pond with several rock compositions representing islands. The upper garden is a dry rock garden featuring three rock “islands.” The first, called Kameshima (the island of the turtle), resembles a turtle swimming in a “lake” of moss. The second, Zazen-seki, is a flat meditation rock that is believed to radiate calm and silence. The third island is the kare-taki, a dry “waterfall” composed of a stairway of flat granite rocks. The moss that now surrounds the rocks and represents water was not part of the original garden plan; it grew several centuries later when the garden was left untended. However, it is now the most famous feature of the garden.
Tenryū-ji
Muso Kokushi built another temple garden at Tenryū-ji, known as the Temple of the Celestial Dragon. This garden appears to have been strongly influenced by Chinese landscape paintings of the Song Dynasty, which feature mountains rising in the mist and suggest great depth and height. The garden at Tenryū-ji has a real pond with water and a dry waterfall of rocks, appearing similar to a Chinese landscape. Saihō-ji and Tenryū-ji show the transition from the Heian style garden toward a more abstract and stylized view of nature.
Ginkaku-ji
The gardens of Ginkaku-ji, also known as the Silver Pavilion, are also attributed to Muso Kokushi. This temple garden includes a traditional pond garden, but it had a new feature for a Japanese garden: an area of raked white gravel with a perfectly shaped mountain of white gravel, resembling Mount Fuji, in the center. The scene is called ginshanada, or “sand of silver and open sea.” This garden feature became known as kogetsudai, or “small mountain facing the moon.” After this garden was built, similar small Mount Fujis made of sand or earth covered with grass appeared in Japanese gardens for centuries afterwards.
Ryōan-ji
The most famous of all zen gardens in Kyoto is Ryōan-ji, built in the late 15th century where, for the first time, the zen garden became purely abstract. The garden is a rectangle of 340 square meters. Placed within it are 15 stones of different sizes, carefully composed in five groups: one group of five stones, two groups of three, and two groups of two stones. The stones are surrounded by white gravel, which is carefully raked every day by the monks. The only vegetation in the garden is some moss around the stones. The garden is meant to be viewed from a seated position on the veranda of the hōjō, the residence of the abbot of the monastery.
Daisen-in
The garden at Daisen-in (1509–1513) took a more literary approach than Ryōan-ji. A “river” of white gravel represents a metaphorical journey through life—beginning with a dry waterfall in the mountains, passing through rapids and rocks, and ending in a tranquil sea of white gravel with two gravel mountains.
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
- Buddhist art in Japan. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Buddhist_art_in_Japan. License : CC BY-SA: Attribution-ShareAlike
- calligraphy. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/calligraphy. License : CC BY-SA: Attribution-ShareAlike
- Koan. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Koan. License : CC BY-SA: Attribution-ShareAlike
- Shubun - Reading in a Bamboo Grove detail. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Shubun_-_Reading_in_a_Bamboo_Grove_detail.jpg. License : Public Domain: No Known Copyright
- Sesshu - Haboku-Sansui. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Sesshu_-_Haboku-Sansui.jpg. License : Public Domain: No Known Copyright
- Japanese rock garden. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Japanese_rock_garden. License : CC BY-SA: Attribution-ShareAlike
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