## Section10.1Function Basics

###### ObjectivesPCC Course Content and Outcome Guide

In Section 9.1 there is a light introduction to functions. This chapter introduces functions more thoroughly, and is independent from Section 9.1.

### Subsection10.1.1Informal Definition of a Function

We are familiar with the $\sqrt{\phantom{x}}$ symbol. This symbol is used to turn numbers into their square roots. Sometimes it's simple to do this on paper or in our heads, and sometimes it helps to have a calculator. We can see some calculations in Table 10.1.2.

 $\sqrt{9}$ ${}=3$ $\sqrt{1/4}$ ${}=1/2$ $\sqrt{2}$ ${}\approx1.41$

The $\sqrt{\phantom{x}}$ symbol represents a process; it's a way for us to turn numbers into other numbers. This idea of having a process for turning numbers into other numbers is the fundamental topic of this chapter.

###### Definition10.1.3Function (Informal Definition)

A function is a process for turning numbers into (potentially) different numbers. The process must be consistent, in that whenever you apply it to some particular number, you always get the same result.

Section 10.5 covers a more technical definition for functions, and gets into some topics that are more appropriate when using that definition. Definition 10.1.3 is so broad that you probably use functions all the time.

###### Example10.1.4

Think about each of these examples, where some process is used for turning one number into another.

• If you convert a person's birth year into their age, you are using a function.

• If you look up the Kelly Blue Book value of a Honda Odyssey based on how old it is, you are using a function.

• If you use the expected guest count for a party to determine how many pizzas you should order, you are using a function.

The process of using $\sqrt{\phantom{x}}$ to change numbers might feel more “mathematical” than these examples. Let's continue thinking about $\sqrt{\phantom{x}}$ for now, since it's a formula-like symbol that we are familiar with. One concern with $\sqrt{\phantom{x}}$ is that although we live in the modern age of computers, this symbol is not found on most keyboards. And yet computers still tend to be capable of producing square roots. Computer technicians write $\operatorname{sqrt}(\phantom{x})$ when they want to compute a square root, as we see in Table 10.1.5.

 $\operatorname{sqrt}(9)$ ${}=3$ $\operatorname{sqrt}(1/4)$ ${}=1/2$ $\operatorname{sqrt}(2)$ ${}\approx1.41$

The parentheses are very important. To see why, try to put yourself in the “mind” of a computer, and look closely at sqrt49. The computer will recognize sqrt and know that it needs to compute a square root. But computers have myopic vision and they might not see the entire number $49\text{.}$ A computer might think that it needs to compute sqrt4 and then append a 9 to the end, which would produce a final result of $29\text{.}$ This is probably not what was intended. And so the purpose of the parentheses in sqrt(49) is to denote exactly what number needs to be operated on.

This demonstrates the standard notation that is used worldwide to write down most functions. By having a standard notation for communicating about functions, people from all corners of the earth can all communicate mathematics with each other more easily, even when they don't speak the same language.

Functions have their own names. We've seen a function named $\operatorname{sqrt}\text{,}$ but any name you can imagine is allowable. In the sciences, it is common to name functions with whole words, like $\operatorname{weight}$ or $\operatorname{health\_index}\text{.}$ In mathematics, we often abbreviate such function names to $w$ or $h\text{.}$ And of course, since the word “function” itself starts with “f,” we will often name a function $f\text{.}$

It's crucial to continue reminding ourselves that functions are processes for changing numbers; they are not numbers themselves. And that means that we have a potential for confusion that we need to stay aware of. In some contexts, the symbol $t$ might represent a variable—a number that is represented by a letter. But in other contexts, $t$ might represent a function—a process for changing numbers into other numbers. By staying conscious of the context of an investigation, we avoid confusion.

Next we need to discuss how we go about using a function's name.

###### Definition10.1.6Function Notation

The standard notation for referring to functions involves giving the function itself a name, and then writing:

\begin{equation*} \begin{matrix} \text{name}\\ \text{of}\\ \text{function} \end{matrix} \left( \begin{matrix} \\ \text{input}\\ \\ \end{matrix} \right) \end{equation*}
###### Example10.1.7

$f(13)$ is pronounced “f of 13.” The word “of” is very important, because it reminds us that $f$ is a process and we are about to apply that process to the input value $13\text{.}$ So $f$ is the function, $13$ is the input, and $f(13)$ is the output we'd get from using $13$ as input.

$f(x)$ is pronounced “f of x.” This is just like the previous example, except that the input is not any specific number. The value of $x$ could be $13$ or any other number. Whatever $x$'s value, $f(x)$ means the corresponding output from the function $f\text{.}$

$\operatorname{BudgetDeficit}(2017)$ is pronounced “BudgetDeficit of 2017.” This is probably about a function that takes a year as input, and gives that year's federal budget deficit as output. The process here of changing a year into a dollar amount might not involve any mathematical formula, but rather looking up information from the Congressional Budget Office's website.

$\operatorname{Celsius}(F)$ is pronounced “Celsius of F.” This is probably about a function that takes a Fahrenheit temperature as input and gives the corresponding Celsius temperature as output. Maybe a formula is used to do this; maybe a chart or some other tool is used to do this. Here, $\operatorname{Celsius}$ is the function, $F$ is the input variable, and $\operatorname{Celsius}(F)$ is the output from the function.

###### Note10.1.8

While a function has a name like $f\text{,}$ and the input to that function often has a variable name like $x\text{,}$ the expression $f(x)$ represents the output of the function. To be clear, $f(x)$ is not a function. Rather, $f$ is a function, and $f(x)$ its output when the number $x$ was used as input.

###### Warning10.1.10Notation Ambiguity

As mentioned earlier, we need to remain conscious of the context of any symbol we are using. It's possible for $f$ to represent a function (a process), but it's also possible for $f$ to represent a variable (a number). Similarly, parentheses might indicate the input of a function, or they might indicate that two numbers need to be multiplied. It's up to our judgment to interpret algebraic expressions in the right context. Consider the expression $a(b)\text{.}$ This could easily mean the output of a function $a$ with input $b\text{.}$ It could also mean that two numbers $a$ and $b$ need to be multiplied. It all depends on the context in which these symbols are being used.

Sometimes it's helpful to think of a function as a machine, as in Figure 10.1.11. This illustrates how complicated functions can be. A number is just a number. But a function has the capacity to take in all kinds of different numbers into it's hopper (feeding tray) as inputs and transform them into their outputs.

### Subsection10.1.2Tables and Graphs

Since functions are potentially complicated, we want ways to understand them more easily. Two basic tools for understanding a function better are tables and graphs.

###### Example10.1.12A Table for the Budget Deficit Function

Consider the function $\operatorname{BudgetDeficit}\text{,}$ that takes a year as its input and outputs the US federal budget deficit for that year. For example, the Congressional Budget Office's website tells us that $\operatorname{BudgetDeficit}(2009)$ is $\1.41$ trillion. If we'd like to understand this function better, we might make a table of all the inputs and outputs we can find. Using the CBO's website (www.cbo.gov/topics/budget), we can put together Table 10.1.13.

###### 94

Suppose that $f$ is the function that gives the total cost (in dollars) of downhill skiing $x$ times during a season with a \$500 season pass. Write a formula for $f\text{.}$

###### 95

Suppose that $f$ is the function that tells you how many dimes are in $x$ dollars. Write a formula for $f\text{.}$

###### 96

The following figure has the graph $y=d(t)\text{,}$ which models a particle’s distance from the starting line in feet, where $t$ stands for time in seconds since timing started.

1. $d(3)=$

2. Interpret the meaning of $d(3)\text{:}$

• A. The particle was $3$ feet away from the starting line $6$ seconds since timing started.

• B. The particle was $6$ feet away from the starting line $3$ seconds since timing started.

• C. In the first $6$ seconds, the particle moved a total of $3$ feet.

• D. In the first $3$ seconds, the particle moved a total of $6$ feet.

3. Solve $d(t)={2}$ for $t\text{.}$ $t=$

4. Interpret the meaning of part c’s solution(s):

• A. The article was $2$ feet from the starting line $9$ seconds since timing started.

• B. The article was $2$ feet from the starting line $1$ seconds since timing started, and again $9$ seconds since timing started.

• C. The article was $2$ feet from the starting line $1$ seconds since timing started, or $9$ seconds since timing started.

• D. The article was $2$ feet from the starting line $1$ seconds since timing started.

###### 97

The following figure has the graph $y=d(t)\text{,}$ which models a particle’s distance from the starting line in feet, where $t$ stands for time in seconds since timing started.

1. $d(4)=$

2. Interpret the meaning of $d(4)\text{:}$

• A. The particle was $4$ feet away from the starting line $10$ seconds since timing started.

• B. In the first $10$ seconds, the particle moved a total of $4$ feet.

• C. The particle was $10$ feet away from the starting line $4$ seconds since timing started.

• D. In the first $4$ seconds, the particle moved a total of $10$ feet.

3. Solve $d(t)={5}$ for $t\text{.}$ $t=$

4. Interpret the meaning of part c’s solution(s):

• A. The article was $5$ feet from the starting line $1$ seconds since timing started.

• B. The article was $5$ feet from the starting line $1$ seconds since timing started, and again $8$ seconds since timing started.

• C. The article was $5$ feet from the starting line $1$ seconds since timing started, or $8$ seconds since timing started.

• D. The article was $5$ feet from the starting line $8$ seconds since timing started.

###### 98

The function $C$ models the the number of customers in a store $t$ hours since the store opened.

 $t$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $C(t)$ $0$ $43$ $81$ $102$ $98$ $83$ $43$ $0$
1. $C(2)=$

2. Interpret the meaning of $C(2)\text{:}$

• A. There were $81$ customers in the store $2$ hours after the store opened.

• B. There were $2$ customers in the store $81$ hours after the store opened.

• C. In $2$ hours since the store opened, the store had an average of $81$ customers per hour.

• D. In $2$ hours since the store opened, there were a total of $81$ customers.

3. Solve $C(t)=43$ for $t\text{.}$ $t=$

4. Interpret the meaning of Part c’s solution(s):

• A. There were $43$ customers in the store $1$ hours after the store opened.

• B. There were $43$ customers in the store either $1$ hours after the store opened, or $6$ hours after the store opened.

• C. There were $43$ customers in the store $6$ hours after the store opened.

• D. There were $43$ customers in the store $1$ hours after the store opened, and again $6$ hours after the store opened.

###### 99

Chicago's average monthly rainfall, $R = f(t)$ inches, is given as a function of the month, $t\text{,}$ where January is $t = 1\text{,}$ in the table below.

 t, month 1 2 3 4 5 6 7 8 R, inches 1.8 1.8 2.7 3.1 3.5 3.7 3.5 3.4

(a) Solve $f(t) = 3.1\text{.}$

$t =$

The solution(s) to $f(t) = 3.1$ can be interpreted as saying

• Chicago's average rainfall in the month of April is 3.1 inches.

• Chicago's average rainfall is greatest in the month of February.

• Chicago's average rainfall increases by 3.1 inches in the month of February.

• Chicago's average rainfall is least in the month of April.

• None of the above

(b) Solve $f(t) = f(5)\text{.}$

$t =$

The solution(s) to $f(t) = f(5)$ can be interpreted as saying

• Chicago's average rainfall is 3.5 inches in the months of May and July.

• Chicago's average rainfall is 3.5 inches in the month of July.

• Chicago's average rainfall is greatest in the month of May.

• Chicago's average rainfall is 3.5 inches in the month of May.

• None of the above

###### 100

Let $f(t)$ denote the number of people eating in a restaurant $t$ minutes after 5 PM. Answer the following questions:

a) Which of the following statements best describes the significance of the expression $f( 4 ) = 19\text{?}$

• There are 19 people eating at 9:00 PM

• Every 4 minutes, 19 more people are eating

• There are 19 people eating at 5:04 PM

• There are 4 people eating at 5:19 PM

• None of the above

b) Which of the following statements best describes the significance of the expression $f(a) = 30\text{?}$

• At 5:30 PM there are $a$ people eating

• Every 30 minutes, the number of people eating has increased by $a$ people

• $a$ hours after 5 PM there are 30 people eating

• $a$ minutes after 5 PM there are 30 people eating

• None of the above

c) Which of the following statements best describes the significance of the expression $f( 30 ) = b\text{?}$

• $b$ hours after 5 PM there are 30 people eating

• Every 30 minutes, the number of people eating has increased by $b$ people

• At 5:30 PM there are $b$ people eating

• $b$ minutes after 5 PM there are 30 people eating

• None of the above

d) Which of the following statements best describes the significance of the expression $n=f(t)\text{?}$

• $n$ hours after 5 PM there are $t$ people eating

• $t$ hours after 5 PM there are $n$ people eating

• Every $t$ minutes, $n$ more people have begun eating

• $n$ minutes after 5 PM there are $t$ people eating

• None of the above

###### 101

Let $s(t)={12t^{2}-2t+300}\text{,}$ where $s$ is the position (in mi) of a car driving on a straight road at time $t$ (in hr). The car’s velocity (in mi/hr) at time $t$ is given by $v(t)={24t-2}\text{.}$

1. Using function notation, express the car’s position after $2.5$ hours. The answer here is not a formula, it’s just something using function notation like f(8).

2. Where is the car then? The answer here is a number with units.

3. Use function notation to express the question, “When is the car going ${65\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut hr}}}\text{?}$” The answer is an equation that uses function notation; something like f(x)=23. You are not being asked to actually solve the equation, just to write down the equation.

4. Where is the car when it is going ${22\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut hr}}}\text{?}$ The answer here is a number with units. You are being asked a question about its position, but have been given information about its speed.

###### 102

Let $s(t)={13t^{2}-3t+300}\text{,}$ where $s$ is the position (in mi) of a car driving on a straight road at time $t$ (in hr). The car’s velocity (in mi/hr) at time $t$ is given by $v(t)={26t-3}\text{.}$

1. Using function notation, express the car’s position after $3.9$ hours. The answer here is not a formula, it’s just something using function notation like f(8).

2. Where is the car then? The answer here is a number with units.

3. Use function notation to express the question, “When is the car going ${61\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut hr}}}\text{?}$” The answer is an equation that uses function notation; something like f(x)=23. You are not being asked to actually solve the equation, just to write down the equation.

4. Where is the car when it is going ${75\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut hr}}}\text{?}$ The answer here is a number with units. You are being asked a question about its position, but have been given information about its speed.

###### 103

Describe your own example of a function that has real context to it. You will need some kind of input variable, like “number of years since 2000” or “weight of the passengers in my car.” You will need a process for using that number to bring about a different kind of number. The process does not need to involve a formula; a verbal description would be great, as would a formula.

Give your function a name. Write the symbol(s) that you would use to represent input. Write the symbol(s) that you would use to represent output.

###### 104

Use the graph of $h$ in the figure to fill in the table.

 $x$ $-2$ $-1$ $0$ $1$ $2$ $h(x)$
1. Evaluate $h(3)-h(0)\text{.}$

2. Evaluate $h(2)-h(-1)\text{.}$

3. Evaluate $2h(-1)\text{.}$

4. Evaluate $h(0)+3\text{.}$

###### 105

Use the given graph of a function $f\text{,}$ along with $a, b, c, d, e\text{,}$ and $h$ to answer the following questions. Some answers are points, and should be entered as ordered pairs. Some answers ask you to solve for $x\text{,}$ so the answer should be in the form x=...

1. What are the coordinates of the point $P\text{?}$

2. What are the coordinates of the point $Q\text{?}$

3. Evaluate $f(b)\text{.}$ (The answer is symbolic, not a specific number.)

4. Solve $f(x)=e$ for $x\text{.}$ (The answer is symbolic, not a specific number.)

5. Suppose $c=f(z)\text{.}$ Solve the equation $z=f(x)$ for $x\text{.}$