Review of Functions

In this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Most of this material will be a review for you, but it serves as a handy reference to remind you of some of the algebraic techniques useful for working with functions.

Functions

defines a relationship between those two sets. A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the input; the element of the second set is called the output. Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.

where the domain is the set of all real numbers and the rule is to square the input. Then, the input

x = 3

Since every nonnegative real number has a real-value square root, every nonnegative number is an element of the range of this function. Since there is no real number with a square that is negative, the negative real numbers are not elements of the range. We conclude that the range is the set of nonnegative real numbers.

The graph of a function is the set of all these points. For example, consider the function

f,

Every function has a domain. However, sometimes a function is described by an equation, as in

f (x) = x^{2},

with no specific domain given. In this case, the domain is taken to be the set of all real numbers

x

is a real number. For example, since any real number can be squared, if no other domain is specified, we consider the domain of

f (x) = x^{2}

to be the set of all real numbers. On the other hand, the square root function

f (x) = \sqrt{x}

the domains are sets with an infinite number of elements. Clearly we cannot list all these elements. When describing a set with an infinite number of elements, it is often helpful to use set-builder or interval notation. When using set-builder notation to describe a subset of all real numbers, denoted

ℝ,

has some property. For example, if we were interested in the set of real numbers that are greater than one but less than five, we could denote this set using set-builder notation by writing

are called the endpoints of this set. If we want to consider the set that includes the endpoints, we would denote this set by writing

We can use similar notation if we want to include one of the endpoints, but not the other. To denote the set of nonnegative real numbers, we would use the set-builder notation

The smallest number in this set is zero, but this set does not have a largest number. Using interval notation, we would use the symbol

\infty,

is not a real number. It is used symbolically here to indicate that this set includes all real numbers greater than or equal to zero. Similarly, if we wanted to describe the set of all nonpositive numbers, we could write

refers to negative infinity, and it indicates that we are including all numbers less than or equal to zero, no matter how small. The set

Some functions are defined using different equations for different parts of their domain. These types of functions are known as piecewise-defined functions. For example, suppose we want to define a function

f

Finding Domain and Range

For each of the following functions, determine the i. domain and ii. range.

$f (x) = {(x - 4)}^{2} + 5$
$f (x) = \sqrt{3 x + 2} - 1$
$f (x) = \frac{3}{x - 2}$

Consider f(x)=(x−4)2+5.
1. Since $f (x) = {(x - 4)}^{2} + 5$
  is a real number for any real number
  $x,$
  the domain of
  $f$
  is the interval
  $(− \infty, \infty) .$
2. Since ${(x - 4)}^{2} \geq 0,$
  we know
  $f (x) = {(x - 4)}^{2} + 5 \geq 5 .$
  Therefore, the range must be a subset of
  ${y \| y \geq 5} .$
  To show that every element in this set is in the range, we need to show that for a given
  $y$
  in that set, there is a real number
  $x$
  such that
  $f (x) = {(x - 4)}^{2} + 5 = y .$
  Solving this equation for
  $x,$
  we see that we need
  $x$
  such that
  
  ${(x - 4)}^{2} = y - 5 .$
  
  This equation is satisfied as long as there exists a real number
  $x$
  such that
  
  $x - 4 = \pm \sqrt{y - 5} .$
  
  Since
  $y \geq 5,$
  the square root is well-defined. We conclude that for
  $x = 4 \pm \sqrt{y - 5}, f (x) = y,$
  and therefore the range is
  ${y \| y \geq 5} .$
Consider f(x)=3x+2−1.
1. To find the domain of $f,$
  we need the expression
  $3 x + 2 \geq 0 .$
  Solving this inequality, we conclude that the domain is
  ${x \| x \geq −2 / 3} .$
2. To find the range of $f,$
  we note that since
  $\sqrt{3 x + 2} \geq 0, f (x) = \sqrt{3 x + 2} - 1 \geq −1 .$
  Therefore, the range of
  $f$
  must be a subset of the set
  ${y \| y \geq −1} .$
  To show that every element in this set is in the range of
  $f,$
  we need to show that for all
  $y$
  in this set, there exists a real number
  $x$
  in the domain such that
  $f (x) = y .$
  Let
  $y \geq −1 .$
  Then,
  $f (x) = y$
  if and only if
  
  $\sqrt{3 x + 2} - 1 = y .$
  
  Solving this equation for
  $x,$
  we see that
  $x$
  must solve the equation
  
  $\sqrt{3 x + 2} = y + 1 .$
  
  Since
  $y \geq −1,$
  such an
  $x$
  could exist. Squaring both sides of this equation, we have
  $3 x + 2 = {(y + 1)}^{2} .$
  
  Therefore, we need
  
  $3 x = {(y + 1)}^{2} - 2,$
  
  which implies
  
  $x = \frac{1}{3} {(y + 1)}^{2} - \frac{2}{3} .$
  
  We just need to verify that
  $x$
  is in the domain of
  $f .$
  Since the domain of
  $f$
  consists of all real numbers greater than or equal to
  $−2 / 3,$
  and
  
  $\frac{1}{3} {(y + 1)}^{2} - \frac{2}{3} \geq - \frac{2}{3},$
  
  there does exist an
  $x$
  in the domain of
  $f .$
  We conclude that the range of
  $f$
  is
  ${y \| y \geq −1} .$
Consider f(x)=3/(x−2).
1. Since $3 / (x - 2)$
  is defined when the denominator is nonzero, the domain is
  ${x \| x \neq 2} .$
2. To find the range of $f,$
  we need to find the values of
  $y$
  such that there exists a real number
  $x$
  in the domain with the property that
  
  $\frac{3}{x - 2} = y .$
  
  Solving this equation for
  $x,$
  we find that
  
  $x = \frac{3}{y} + 2 .$
  
  Therefore, as long as
  $y \neq 0,$
  there exists a real number
  $x$
  in the domain such that
  $f (x) = y .$
  Thus, the range is
  ${y \| y \neq 0} .$

Representing Functions

We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula.

Tables

Functions described using a table of values arise frequently in real-world applications. Consider the following simple example. We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable

x

hours after midnight, measured in degrees Fahrenheit. We record our data in [link].

We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it.

Graphs

Temperature as a Function of Time of Day
Hours after Midnight	Temperature $(° F)$	Hours after Midnight	Temperature $(° F)$
0	58	12	84
1	54	13	85
2	53	14	85
3	52	15	83
4	52	16	82
5	55	17	80
6	60	18	77
7	64	19	74
8	72	20	69
9	75	21	65
10	78	22	60
11	80	23	58

described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in [link] can give us a better idea of their fluctuation throughout the day. [link] shows the plot of the temperature function.

From the points plotted on the graph in [link], we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table. In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern, as we can see in [link].

Algebraic Formulas

Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications. For example, the area of a circle of radius

r

When an object is thrown upward from the ground with an initial velocity

v_{0}

ft/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula

s (t) = −16 t^{2} + v_{0} t .

Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.

is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of

f

consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin.

When creating a table of inputs and outputs, we typically check to determine whether zero is an output. Those values of

x

are called the zeros of a function. For example, the zeros of

f (x) = x^{2} - 4

-axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the x-axis, or it may intersect multiple (or even infinitely many) times.

Since a function has exactly one output for each input, the graph of a function can have, at most, one

y

We can use this test to determine whether a set of plotted points represents the graph of a function ([link]).

Height $s$ as a Function of Time $t$
$t$	$0$	$0.5$	$1$	$1.5$	$2$	$2.5$
$s (t)$	$100$	$96$	$84$	$64$	$36$	$0$

is getting larger. A function with this property is said to be decreasing. On the other hand, for the function

f (x) = \sqrt{x + 3} + 1

are getting larger. A function with this property is said to be increasing. It is important to note, however, that a function can be increasing on some interval or intervals and decreasing over a different interval or intervals. For example, using our temperature function in [link], we can see that the function is decreasing on the interval

(0, 4),

We make the idea of a function increasing or decreasing over a particular interval more precise in the next definition.

Combining Functions

Now that we have reviewed the basic characteristics of functions, we can see what happens to these properties when we combine functions in different ways, using basic mathematical operations to create new functions. For example, if the cost for a company to manufacture

x

Alternatively, we can create a new function by composing two functions. For example, given the functions

f (x) = x^{2}

Combining Functions with Mathematical Operators

To combine functions using mathematical operators, we simply write the functions with the operator and simplify. Given two functions

f

Function Composition

When we compose functions, we take a function of a function. For example, suppose the temperature

T

to heat or cool a building for 1 hour, can be described as a function of the temperature

T .

Combining these two functions, we can describe the cost of heating or cooling a building as a function of time by evaluating

C (T (t)) .

This new function is called a composite function. We note that since cost is a function of temperature and temperature is a function of time, it makes sense to define this new function

(C \circ T) (t) .

This tells us, in general terms, that the order in which we compose functions matters.

Symmetry of Functions

The graphs of certain functions have symmetry properties that help us understand the function and the shape of its graph. For example, consider the function

f (x) = x^{4} - 2 x^{2} - 3

shown in [link](a). If we take the part of the curve that lies to the right of the y-axis and flip it over the y-axis, it lays exactly on top of the curve to the left of the y-axis. In this case, we say the function has symmetry about the y-axis. On the other hand, consider the function

f (x) = x^{3} - 4 x

about the origin, the new graph will look exactly the same. In this case, we say the function has symmetry about the origin.

If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function

f

is an even function, which has symmetry about the y-axis. For example,

f (x) = x^{2}

is an odd function, which has symmetry about the origin. For example,

f (x) = x^{3}

One symmetric function that arises frequently is the absolute value function, written as

\| x \| .

Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if

x < 0,

Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if

x = 0,

We conclude that the range of the absolute value function is

{y \| y \geq 0} .

In [link], we see that the absolute value function is symmetric about the y-axis and is therefore an even function.

Key Concepts

Key Equations

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

x

y

x

y

| {: valign=”top”}|———- | −3 | 9 | 1 | 1 | {: valign=”top”}| −2 | 4 | 2 | 4 | {: valign=”top”}| −1 | 1 | 3 | 9 | {: valign=”top”}| 0 | 0 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “-3; -2; -1; 0; 1; 2; 3”. The second column is labeled “y” and the values are “9; 4; 1; 0; 1; 4; 9”.” data-label=””}

a. Domain = ${−3, −2, −1, 0, 1, 2, 3},$

range = ${0, 1, 4, 9}$

b. Yes, a function

x

y

x

y

| {: valign=”top”}|———- | −3 | −2 | 1 | 1 | {: valign=”top”}| −2 | −8 | 2 | 8 | {: valign=”top”}| −1 | −1 | 3 | −2 | {: valign=”top”}| 0 | 0 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “-3; -2; -1; 0; 1; 2; 3”. The second column is labeled “y” and the values are “-3; -2; -1; 0; 1; 2; 3”.” data-label=””}

x

y

x

y

| {: valign=”top”}|———- | 1 | −3 | 1 | 1 | {: valign=”top”}| 2 | −2 | 2 | 2 | {: valign=”top”}| 3 | −1 | 3 | 3 | {: valign=”top”}| 0 | 0 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “1; 2; 3; 0; 1; 2; 3”. The second column is labeled “y” and the values are “-3; -2; -1; 0; 1; 2; 3”.” data-label=””}

a. Domain = ${0, 1, 2, 3},$

range = ${−3, −2, −1, 0, 1, 2, 3}$

b. No, not a function

x

y

x

y

| {: valign=”top”}|———- | 1 | 1 | 5 | 1 | {: valign=”top”}| 2 | 1 | 6 | 1 | {: valign=”top”}| 3 | 1 | 7 | 1 | {: valign=”top”}| 4 | 1 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “1; 2; 3; 4; 5; 6; 7”. The second column is labeled “y” and the values are “1; 1; 1; 1; 1; 1; 1”.” data-label=””}

x

y

x

y

| {: valign=”top”}|———- | 3 | 3 | 15 | 1 | {: valign=”top”}| 5 | 2 | 21 | 2 | {: valign=”top”}| 8 | 1 | 33 | 3 | {: valign=”top”}| 10 | 0 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “3; 5; 8; 10; 15; 21; 33”. The second column is labeled “y” and the values are “3; 2; 1; 0; 1; 2; 3”.” data-label=””}

a. Domain = ${3, 5, 8, 10, 15, 21, 33},$

range = ${0, 1, 2, 3}$

b. Yes, a function

x

y

x

y

| {: valign=”top”}|———- | −7 | 11 | 1 | −2 | {: valign=”top”}| −2 | 5 | 3 | 4 | {: valign=”top”}| −2 | 1 | 6 | 11 | {: valign=”top”}| 0 | −1 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “-7; -2; -2; 0; 1; 3; 6”. The second column is labeled “y” and the values are “11; 5; 1; -1; -2; 4; 11”.” data-label=””}

For the following exercises, find the values for each function, if they exist, then simplify.

a. $f (0)$

b. $f (1)$

c. $f (3)$

d. $f (− x)$

e. $f (a)$

f. $f (a + h)$

f (x) = 5 x - 2

a. $−2$

b. 3 c. 13 d. $−5 x - 2$

e. $5 a - 2$

f. $5 a + 5 h - 2$

f (x) = 4 x^{2} - 3 x + 1

f (x) = \frac{2}{x}

a. Undefined b. 2 c. $\frac{2}{3}$

d. $- \frac{2}{x}$

e $\frac{2}{a}$

f. $\frac{2}{a + h}$

f (x) = \| x - 7 \| + 8

f (x) = \sqrt{6 x + 5}

a. $\sqrt{5}$

b. $\sqrt{11}$

c. $\sqrt{23}$

d. $\sqrt{−6 x + 5}$

e. $\sqrt{6 a + 5}$

f. $\sqrt{6 a + 6 h + 5}$

f (x) = \frac{x - 2}{3 x + 7}

f (x) = 9

a. 9 b. 9 c. 9 d. 9 e. 9 f. 9

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.

f (x) = \frac{x}{x^{2} - 16}

g (x) = \sqrt{8 x - 1}

x \geq \frac{1}{8}; y \geq 0; x = \frac{1}{8};

no y-intercept

h (x) = \frac{3}{x^{2} + 4}

f (x) = −1 + \sqrt{x + 2}

x \geq −2; y \geq −1; x = −1; y = −1 + \sqrt{2}

f (x) = \frac{1}{\sqrt{x - 9}}

g (x) = \frac{3}{x - 4}

x \neq 4; y \neq 0;

no x-intercept; $y = - \frac{3}{4}$

f (x) = 4 \| x + 5 \|

g (x) = \sqrt{\frac{7}{x - 5}}

x > 5; y > 0;

no intercepts

For the following exercises, set up a table to sketch the graph of each function using the following values: $x = −3, −2, −1, 0, 1, 2, 3 .$

f (x) = x^{2} + 1

x

y

x

y

| {: valign=”top”}|———- | −3 | 10 | 1 | 2 | {: valign=”top”}| −2 | 5 | 2 | 5 | {: valign=”top”}| −1 | 2 | 3 | 10 | {: valign=”top”}| 0 | 1 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “-3; -2; -1; 0; 1; 2; 3”. The second column is labeled “y” and the values are “10; 5; 2; 1; 2; 5; 10”.” data-label=””}

f (x) = 3 x - 6

x

y

x

y

| {: valign=”top”}|———- | −3 | −15 | 1 | −3 | {: valign=”top”}| −2 | −12 | 2 | 0 | {: valign=”top”}| −1 | −9 | 3 | 3 | {: valign=”top”}| 0 | −6 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “-3; -2; -1; 0; 1; 2; 3”. The second column is labeled “y” and the values are “-15; -12; -9; -6; -3; 0; 3”.” data-label=””}

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -3 to 3. The graph is of the function “f(x) = 3x - 6”, which is an increasing straight line. The function has an x intercept at (2, 0) and the y intercept is not shown.

f (x) = \frac{1}{2} x + 1

x

y

x

y


{: valign=”top”}	———-
−3	$- \frac{1}{2}$

1	$\frac{3}{2}$


{: valign=”top”}	−2	0	2	2
{: valign=”top”}	−1	$\frac{1}{2}$

3	$\frac{5}{2}$

| {: valign=”top”}| 0 | 1 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “-3; -2; -1; 0; 1; 2; 3”. The second column is labeled “y” and the values are “-(1/2); 0; (1/2); 1; (3/2); 2; (5/2)”.” data-label=””}

f (x) = 2 \| x \|

x

y

x

y

| {: valign=”top”}|———- | −3 | 6 | 1 | 2 | {: valign=”top”}| −2 | 4 | 2 | 4 | {: valign=”top”}| −1 | 2 | 3 | 6 | {: valign=”top”}| 0 | 0 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “-3; -2; -1; 0; 1; 2; 3”. The second column is labeled “y” and the values are “6; 4; 2; 0; 2; 4; 6”.” data-label=””}

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -2 to 6. The graph is of the function “f(x) = 2 times the absolute value of x”. The function decreases in a straight line until it hits the origin, then begins to increase in a straight line. The function x intercept and y intercept are at the origin.

f (x) = − x^{2}

x

y

x

y

| {: valign=”top”}|———- | −3 | −9 | 1 | −1 | {: valign=”top”}| −2 | −4 | 2 | −4 | {: valign=”top”}| −1 | −1 | 3 | −9 | {: valign=”top”}| 0 | 0 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “-3; -2; -1; 0; 1; 2; 3”. The second column is labeled “y” and the values are “-9; -4; -1; 0; -1; -4; -9”.” data-label=””}

f (x) = x^{3}

x

y

x

y

| {: valign=”top”}|———- | −3 | −27 | 1 | 1 | {: valign=”top”}| −2 | −8 | 2 | 8 | {: valign=”top”}| −1 | −1 | 3 | 27 | {: valign=”top”}| 0 | 0 | | | {: valign=”top”}{: .unnumbered summary=”A table with 7 rows and 2 columns. The first column is labeled “x” and has the values “-3; -2; -1; 0; 1; 2; 3”. The second column is labeled “y” and the values are “-27; -8; -1; 0; 1; 8; 27”.” data-label=””}

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -27 to 27. The graph is of the function “f(x) = x cubed”. The curved function increases until it hits the origin, where it levels out and then becomes even. After the origin the graph begins to increase again. The x intercept and y intercept are both at the origin.

For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph:

Domain and range
$x$
-intercept, if any (estimate where necessary)
$y$
-Intercept, if any (estimate where necessary)
The intervals for which the function is increasing
The intervals for which the function is decreasing
The intervals for which the function is constant
Symmetry about any axis and/or the origin
Whether the function is even, odd, or neither

![An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is circle, with x intercepts at (-1, 0) and (1, 0) and y intercepts at (0, 1) and (0, -1).](/calculus-book/resources/CNX_Calc_Figure_01_01_207.jpg)

![An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is curved. The relation decreases until it hits the point (-1, 0), then increases until it hits the point (0, 1), then decreases until it hits the point (1, 0), then increases again.](/calculus-book/resources/CNX_Calc_Figure_01_01_208.jpg)

Function; a. Domain: all real numbers, range: $y \geq 0$

b. $x = \pm 1$

c. $y = 1$

d. $−1 < x < 0$

and $1 < x < \infty$

e. $− \infty < x < - 1$

and $0 < x < 1$

f. Not constant g. y-axis h. Even

![An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a parabola. The curved relation increases until it hits the point (2, 3), then begins to decrease. The approximate x intercepts are at (0.3, 0) and (3.7, 0) and the y intercept is is (-1, 0).](/calculus-book/resources/CNX_Calc_Figure_01_01_209.jpg)

![An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is curved. The curved relation increases the entire time. The x intercept and y intercept are both at the origin.](/calculus-book/resources/CNX_Calc_Figure_01_01_217.jpg)

Function; a. Domain: all real numbers, range: $−1.5 \leq y \leq 1.5$

b. $x = 0$

c. $y = 0$

d. $all real numbers$

e. None f. Not constant g. Origin h. Odd

![An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a sideways parabola, opening up to the right. The x intercept and y intercept are both at the origin and the relation has no points to the left of the y axis. The relation includes the points (1, -1) and (1, 1)](/calculus-book/resources/CNX_Calc_Figure_01_01_211.jpg)

![An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a horizontal line until the point (-2, -2), then it begins increasing in a straight line until the point (2, 2). After these points, the relation becomes a horizontal line again. The x intercept and y intercept are both at the origin.](/calculus-book/resources/CNX_Calc_Figure_01_01_212.jpg)

Function; a. Domain: $− \infty < x < \infty,$

range: $−2 \leq y \leq 2$

b. $x = 0$

c. $y = 0$

d. $−2 < x < 2$

e. Not decreasing f. $− \infty < x < - 2$

and $2 < x < \infty$

g. Origin h. Odd

![An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a horizontal line until the origin, then it begins increasing in a straight line. The x intercept and y intercept are both at the origin and there are no points below the x axis.](/calculus-book/resources/CNX_Calc_Figure_01_01_213.jpg)

![An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that starts at the point (-4, 4) and is a horizontal line until the point (0, 4), then it begins decreasing in a curved line until it hits the point (4, -4), where the graph ends. The x intercept is approximately at the point (1.2, 0) and y intercept is at the point (0, 4).](/calculus-book/resources/CNX_Calc_Figure_01_01_214.jpg)

Function; a. Domain: $−4 \leq x \leq 4,$

range: $−4 \leq y \leq 4$

b. $x = 1.2$

c. $y = 4$

d. Not increasing e. $0 < x < 4$

f. $−4 < x < 0$

g. No Symmetry h. Neither

For the following exercises, for each pair of functions, find a. $f + g$

b. $f - g$

c. $f \cdot g$

d. $f / g .$

Determine the domain of each of these new functions.

f (x) = 3 x + 4, g (x) = x - 2

f (x) = x - 8, g (x) = 5 x^{2}

a. $5 x^{2} + x - 8;$

all real numbers b. $−5 x^{2} + x - 8;$

all real numbers c. $5 x^{3} - 40 x^{2};$

all real numbers d. $\frac{x - 8}{5 x^{2}}; x \neq 0$

f (x) = 3 x^{2} + 4 x + 1, g (x) = x + 1

f (x) = 9 - x^{2}, g (x) = x^{2} - 2 x - 3

a. $−2 x + 6;$

all real numbers b. $−2 x^{2} + 2 x + 12;$

all real numbers c. $− x^{4} + 2 x^{3} + 12 x^{2} - 18 x - 27;$

all real numbers d. $- \frac{x + 3}{x + 1}; x \neq − 1, 3$

f (x) = \sqrt{x}, g (x) = x - 2

f (x) = 6 + \frac{1}{x}, g (x) = \frac{1}{x}

a. $6 + \frac{2}{x}; x \neq 0$

b. 6; $x \neq 0$

c. $\frac{6}{x} + \frac{1}{x^{2}}; x \neq 0$

d. $6 x + 1; x \neq 0$

For the following exercises, for each pair of functions, find a. $(f \circ g) (x)$

and b. $(g \circ f) (x)$

Simplify the results. Find the domain of each of the results.

f (x) = 3 x, g (x) = x + 5

f (x) = x + 4, g (x) = 4 x - 1

a. $4 x + 3;$

all real numbers b. $4 x + 15;$

all real numbers

f (x) = 2 x + 4, g (x) = x^{2} - 2

f (x) = x^{2} + 7, g (x) = x^{2} - 3

a. $x^{4} - 6 x^{2} + 16;$

all real numbers b. $x^{4} + 14 x^{2} + 46;$

all real numbers

f (x) = \sqrt{x}, g (x) = x + 9

f (x) = \frac{3}{2 x + 1}, g (x) = \frac{2}{x}

a. $\frac{3 x}{4 + x}; x \neq 0, −4$

b. $\frac{4 x + 2}{3}; x \neq - \frac{1}{2}$

f (x) = \| x + 1 \|, g (x) = x^{2} + x - 4

The table below lists the NBA championship winners for the years 2001 to 2012.

| Year | Winner | {: valign=”top”}|———- | 2001 | LA Lakers | {: valign=”top”}| 2002 | LA Lakers | {: valign=”top”}| 2003 | San Antonio Spurs | {: valign=”top”}| 2004 | Detroit Pistons | {: valign=”top”}| 2005 | San Antonio Spurs | {: valign=”top”}| 2006 | Miami Heat | {: valign=”top”}| 2007 | San Antonio Spurs | {: valign=”top”}| 2008 | Boston Celtics | {: valign=”top”}| 2009 | LA Lakers | {: valign=”top”}| 2010 | LA Lakers | {: valign=”top”}| 2011 | Dallas Mavericks | {: valign=”top”}| 2012 | Miami Heat | {: valign=”top”}{: .unnumbered summary=”A table with 12 rows and 2 columns. The first column is labeled “year” and has the values “2001; 2002; 2003; 2004; 2005; 2006; 2007; 2008; 2009; 2010; 2011; 2012”. The second column is labeled “winner” and the values are “LA Lakers; LA Lakers; San Antonio Spurs; Detroit Pistons; San Antonio Spurs; Miami Heat; San Antonio Spurs; Boston Celtics; LA Lakers; LA Lakers; Dallas Mavericks; Miami Heat”.” data-label=””}

Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not.
Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not.

a. Yes, because there is only one winner for each year. b. No, because there are three teams that won more than once during the years 2001 to 2012.

[T] The area $A$

of a square depends on the length of the side $s .$

Write a function $A (s)$
for the area of a square.
Find and interpret $A (6.5) .$
Find the exact and the two-significant-digit approximation to the length of the sides of a square with area 56 square units.

[T] The volume of a cube depends on the length of the sides $s .$

Write a function $V (s)$
for the area of a square.
Find and interpret $V (11.8) .$

a. $V (s) = s^{3}$

b. $V (11.8) \approx 1643;$

a cube of side length 11.8 each has a volume of approximately 1643 cubic units.

[T] A rental car company rents cars for a flat fee of $20 and an hourly charge of $10.25. Therefore, the total cost $C$

to rent a car is a function of the hours $t$

the car is rented plus the flat fee.

Write the formula for the function that models this situation.
Find the total cost to rent a car for 2 days and 7 hours.
Determine how long the car was rented if the bill is $432.73.

[T] A vehicle has a 20-gal tank and gets 15 mpg. The number of miles N that can be driven depends on the amount of gas x in the tank.

Write a formula that models this situation.
Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) 3/4 of a tank of gas.
Determine the domain and range of the function.
Determine how many times the driver had to stop for gas if she has driven a total of 578 mi.

a. $N (x) = 15 x$

b. i. $N (20) = 15 (20) = 300;$

therefore, the vehicle can travel 300 mi on a full tank of gas. Ii. $N (15) = 225;$

therefore, the vehicle can travel 225 mi on 3/4 of a tank of gas. c. Domain: $0 \leq x \leq 20;$

range: $[0, 300]$

d. The driver had to stop at least once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.

[T] The volume V of a sphere depends on the length of its radius as $V = (4 / 3) π r^{3} .$

Because Earth is not a perfect sphere, we can use the mean radius when measuring from the center to its surface. The mean radius is the average distance from the physical center to the surface, based on a large number of samples. Find the volume of Earth with mean radius $6.371 \times 10^{6}$

[T] A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by $r (t) = 6 - [5 / (t^{2} + 1)],$

where t is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture.

Express the area of the bacteria as a function of time.
Find the exact and approximate area of the bacterial culture in 3 hours.
Express the circumference of the bacteria as a function of time.
Find the exact and approximate circumference of the bacteria in 3 hours.

a. $A (t) = A (r (t)) = π \cdot {(6 - \frac{5}{t^{2} + 1})}^{2}$

b. Exact: $\frac{121 π}{4};$

approximately 95 cm² c. $C (t) = C (r (t)) = 2 π (6 - \frac{5}{t^{2} + 1})$

d. Exact: $11 π;$

approximately 35 cm

[T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function $E (x) = 0.79 x,$

where x is the number of U.S. dollars and $E (x)$

is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is $D (x) = 1.245 x,$

where x is the number of Euros and $D (x)$

is the equivalent number of U.S. dollars.

Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process?
Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she converted an extra $200 when she arrived in Paris.

[T] The manager at a skateboard shop pays his workers a monthly salary S of $750 plus a commission of $8.50 for each skateboard they sell.

Write a function $y = S (x)$
that models a worker’s monthly salary based on the number of skateboards x he or she sells.
Find the approximate monthly salary when a worker sells 25, 40, or 55 skateboards.
Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of $1400. (Hint: Find the intersection of the function and the line $y = 1400 .)$

An image of a graph. The y axis runs from 0 to 1800 and the x axis runs from 0 to 100. The graph is of the function “S(x) = 8.5x + 750”, which is a increasing straight line. The function has a y intercept at (0, 750) and the x intercept is not shown.

a. $S (x) = 8.5 x + 750$

b. $962.50, $1090, $1217.50 c. 77 skateboards

[T] Use a graphing calculator to graph the half-circle $y = \sqrt{25 - {(x - 4)}^{2}} .$

Then, use the INTERCEPT feature to find the value of both the $x$

and $y$

-intercepts.

An image of a graph. The y axis runs from -6 to 6 and the x axis runs from -1 to 10. The graph is of the function that is a semi-circle (the top half of a circle). The function has the begins at the point (-1, 0), runs through the point (0, 3), has maximum at the point (4, 5), and ends at the point (9, 0). None of these points are labeled, they are just for reference.

Hours after Midnight	Temperature $(° F)$	Hours after Midnight	Temperature $(° F)$
0	58	12	84
1	54	13	85
2	53	14	85
3	52	15	83
4	52	16	82
5	55	17	80
6	60	18	77
7	64	19	74
8	72	20	69
9	75	21	65
10	78	22	60
11	80	23	58

Hours after Midnight	Temperature $(° F)$	Hours after Midnight	Temperature $(° F)$
0	58	12	84
1	54	13	85
2	53	14	85
3	52	15	83
4	52	16	82
5	55	17	80
6	60	18	77
7	64	19	74
8	72	20	69
9	75	21	65
10	78	22	60
11	80	23	58