🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning

Precalculus 7th

David Cohen, Theodore B. Lee, David Sklar

Chapter 3

Functions

Educators


Problem 1

Let $g$ denote the minimum wage function represented in Table. $$\begin{array}{|c|c|}\hline\text { Year } & \text { Minimum Wage (dollars) } \\\hline 1975 & 2.00 \\1980 & 3.10 \\1985 & 3.35 \\1990 & 3.80 \\1995 & 4.25 \\2000 & 5.15 \\2005 & 5.15 \\2010 & 7.25 \\\hline\end{array}$$
(a) Find $g(1975)$
(b) Find $g(1995)-g(1975)$ and interpret the result.

CC
Charles C.
Numerade Educator

Problem 2

Let $f$ denote the function represented in Table.
$$\begin{array}{|c|c|}\hline\hline \text { Movie Star } & \text { Original Name } \\\hline \text { Kate Capshaw } & \text { Kathleen Sue Nail } \\\text { Tom Cruise } & \text { Thomas Cruise Mapother, IV } \\\text { Demi Moore } & \text { Demetria Guynes } \\\text { Kevin Spacey } & \text { Kevin Fowler } \\\text { Arnold Schwarzenegger } & \text { Arnold Schwarzenegger }\end{array}$$
(a) Find $f($ Kate Capshaw)
(b) For which input $x$ are $x$ and $f(x)$ identical?

CC
Charles C.
Numerade Educator

Problem 3

Let $h$ denote the function represented in Table. $$\begin{array}{|c|c|}\hline\hline\text { Planet } & \text { Number of Moons } \\\hline \text { Mercury } & 0 \\\text { Venus } & 0 \\\text { Earth } & 1 \\\text { Mars } & 2 \\\text { Jupiter } & 16 \\\text { Saturn } & 18 \\\text { Uranus } & 18 \\\text { Neptune } & 8\end{array}$$
(a) Is it the domain or the range of $h$ that consists of real numbers?
(b) Find $h(\text { Mars })$
(c) Which is larger: $h$ (Neptune) or $h$ (Earth)? Interpret the result (using a complete sentence).

CC
Charles C.
Numerade Educator

Problem 4

Again let $h$ denote the function represented in Table 2 .
$$\begin{array}{|c|c|}\hline\hline\text { Planet } & \text { Number of Moons } \\\hline \text { Mercury } & 0 \\\text { Venus } & 0 \\\text { Earth } & 1 \\\text { Mars } & 2 \\\text { Jupiter } & 16 \\\text { Saturn } & 18 \\\text { Uranus } & 18 \\\text { Neptune } & 8\end{array}$$
(a) List those inputs $x$ for which $h(x)=1$.
(b) For which real number $t$ will it be true that liJupiter $)+t=h(\text { Saturn }) ?$

CC
Charles C.
Numerade Educator

Problem 5

Two sets $A$ and $B$ are defined as follows:
$$A=\{x, y, z\} ; B=\{1,2,3\}$$
(a) Which of the rules displayed in the figures represent functions from $A$ to $B$ ?
(b) For each rule that does represent a function, specify the range.
(FIGURES CAN'T COPY)

CC
Charles C.
Numerade Educator

Problem 6

Two sets $A$ and $B$ are defined as follows:
$$A=\{x, y, z\} ; B=\{1,2,3\}$$
(a) Which of the rules displayed in the figures represent functions from $A$ to $B$ ?
(b) For each rule that does represent a function, specify the range.
(FIGURES CAN'T COPY)

CC
Charles C.
Numerade Educator

Problem 7

Two sets $D$ and $C$ are defined as follows:
$$D=\{a, b\} ; C=\{i, j, k\}$$
(a) Which of the rules displayed in the figures represent functions from $D$ to $C$ ?
(b) For each rule that represents a function, specify the range.
(FIGURES CAN'T COPY)

CC
Charles C.
Numerade Educator

Problem 8

Two sets $D$ and $C$ are defined as follows:
$$D=\{a, b\} ; C=\{i, j, k\}$$
(a) Which of the rules displayed in the figures represent functions from $D$ to $C$ ?
(b) For each rule that represents a function, specify the range.
(FIGURES CAN'T COPY)

CC
Charles C.
Numerade Educator

Problem 9

Determine the domain of each function.
(a) $y=-5 x+1$
(b) $y=1 /(-5 x+1)$
(c) $y=\sqrt{-5 x+1}$
(d) $y=\sqrt[3]{-5 x+1}$

CC
Charles C.
Numerade Educator

Problem 10

Determine the domain of each function.
(a) $s=3 t+12$
(b) $s=1 /(3 t+12)$
(c) $s=\sqrt{3 t+12}$
(d) $s=\sqrt[3]{3 t+12}$

CC
Charles C.
Numerade Educator

Problem 11

Determine the domain of each function.
(a) $f(x)=x^{2}-9$
(b) $g(x)=1 /\left(x^{2}-9\right)$
(c) $h(x)=\sqrt{x^{2}-9}$
(d) $k(x)=\sqrt[3]{x^{2}-9}$

CC
Charles C.
Numerade Educator

Problem 12

Determine the domain of each function.
(a) $F(t)=t^{2}+4 t$
(b) $G(t)=1 /\left(t^{2}+4 t\right)$
(c) $H(t)=\sqrt{t^{2}+4 t}$
(d) $K(t)=\sqrt[3]{t^{2}+4 t}$

CC
Charles C.
Numerade Educator

Problem 13

Determine the domain of each function.
(a) $f(t)=t^{2}-8 t+15$
(b) $g(t)=1 /\left(t^{2}-8 t+15\right)$
(c) $h(t)=\sqrt{t^{2}-8 t+15}$
(d) $k(t)=\sqrt[3]{t^{2}-8 t+15}$

CC
Charles C.
Numerade Educator

Problem 14

Determine the domain of each function.
(a) $F(x)=2 x^{2}+x-6$
(b) $G(x)=1 /\left(2 x^{2}+x-6\right)$
(c) $H(x)=\sqrt{2 x^{2}+x-6}$
(d) $K(x)=\sqrt[3]{2 x^{2}+x-6}$

CC
Charles C.
Numerade Educator

Problem 15

Determine the domain of each function.
(a) $f(x)=(x-2) /(2 x+6)$
(b) $g(x)=\sqrt{(x-2) /(2 x+6)}$
(c) $h(x)=\sqrt[3]{(x-2) /(2 x+6)}$

CC
Charles C.
Numerade Educator

Problem 16

Determine the domain of each function.
(a) $F(t)=(3 t-4) /(7-2 t)$
(b) $G(t)=\sqrt{(3 t-4) /(7-2 t)}$
(c) $H(t)=\sqrt[3]{(3 t-4) /(7-2 t)}$

CC
Charles C.
Numerade Educator

Problem 17

Determine the domain and the range of each function.
$$y=4 x-5$$

CC
Charles C.
Numerade Educator

Problem 18

Determine the domain and the range of each function.
$$y=125-12 x$$

CC
Charles C.
Numerade Educator

Problem 19

Determine the domain and the range of each function.
$$y=4 x^{3}-5$$

CC
Charles C.
Numerade Educator

Problem 20

Determine the domain and the range of each function.
$$y=125-12 x^{3}$$

CC
Charles C.
Numerade Educator

Problem 21

Determine the domain and the range of each function.
$$g(x)=\frac{4 x-20}{3 x-18}$$

CC
Charles C.
Numerade Educator

Problem 22

Determine the domain and the range of each function.
$$f(x)=\frac{1-x}{x}$$

CC
Charles C.
Numerade Educator

Problem 23

Determine the domain and the range of each function.
(a) $f(x)=\frac{x+3}{x-5}$
(b) $F(x)=\frac{x^{3}+3}{x^{3}-5}$

CC
Charles C.
Numerade Educator

Problem 24

Determine the domain and the range of each function.
(a) $g(x)=\frac{2 x-7}{3 x+24}$
(b) $G(x)=\frac{2 x^{3}-7}{3 x^{3}+24}$

CC
Charles C.
Numerade Educator

Problem 25

Determine the domain and the range of each function.
$$s=t^{2}+4$$

CC
Charles C.
Numerade Educator

Problem 26

Determine the domain and the range of each function.
$$s=2 t^{2}-10$$

CC
Charles C.
Numerade Educator

Problem 27

Each of the following rules defines a function with domain the set of all real numbers. Express each rule in the form of an equation.
(a) For each real number, subtract 3 and then square the result.
(b) For each real number, compute its square and then subtract
3 from the result.
(c) For each real number, multiply it by 3 and then square the result.
(d) For each real number, compute its square and then multiply the result by 3.

CC
Charles C.
Numerade Educator

Problem 28

Each of the following rules defines a function with domain equal to the set of all real numbers. Express each rule in words.
(a) $y=2 x^{3}+1$
(b) $y=2(x+1)^{3}$
(c) $y=(2 x+1)^{3}$
(d) $y=(2 x)^{3}+1$

CC
Charles C.
Numerade Educator

Problem 29

Let $f(x)=x^{2}-3 x+1 .$ Compute the following.
(a) $f(1)$
(b) $f(0)$
(c) $f(-1)$
(d) $f(3 / 2)$
(e) $f(z)$
(f) $f(x+1)$
(g) $f(a+1)$
(h) $f(-x)$
(i) $|f(1)|$
(j) $f(\sqrt{3})$
(k) $f(1+\sqrt{2})$
(1) $|1-f(2)|$

CC
Charles C.
Numerade Educator

Problem 30

Let $H(x)=1-x+x^{2}-x^{3}$.
(a) Which number is larger, $H(0)$ or $H(1) ?$
(b) Find $H\left(\frac{1}{2}\right) .$ Does $H\left(\frac{1}{2}\right)+H\left(\frac{1}{2}\right)=H(1) ?$

CC
Charles C.
Numerade Educator

Problem 31

Let $f(x)=3 x^{2} .$ Find the following.
(a) $f(2 x)$
(b) $2 f(x)$
(c) $f\left(x^{2}\right)$
(d) $[f(x)]^{2}$
(e) $f(x / 2)$
(f) $f(x) / 2$
For checking: No two answers are the same.

CC
Charles C.
Numerade Educator

Problem 32

Let $f(x)=4-3 x$. Find the following.
(a) $f(2)$
(b) $f(-3)$
(c) $f(2)+f(-3)$
(d) $f(2+3)$
(e) $f(2 x)$
(f) $2 f(x)$
(g) $f\left(x^{2}\right)$
(h) $f(1 / x)$
(i) $f[f(x)]$
(j) $x^{2} f(x)$
(k) $1 / f(x)$
(1) $f(-x)$ $(m)-f(x)$
(n) $-f(-x)$

CC
Charles C.
Numerade Educator

Problem 33

Let $H(x)=1-2 x^{2} .$ Find the following.
(a) $H(\sqrt{2})$
(b) $H(5 / 6)$
(c) $H(x+1)$
(d) $H(x+h)$

CC
Charles C.
Numerade Educator

Problem 34

(a) If $f(x)=2 x+1,$ does $f(3+1)=f(3)+f(1) ?$
(b) If $f(x)=2 x,$ does $f(3+1)=f(3)+f(1) ?$
(c) If $f(x)=\sqrt{x},$ does $f(3+1)=f(3)+f(1) ?$

CC
Charles C.
Numerade Educator

Problem 35

Let $g(x)=2,$ for all $x .$ Find each output.
(a) $g(0)$
(b) $g(5)$
(c) $g(x+h)$

CC
Charles C.
Numerade Educator

Problem 36

Let $g(t)=|t-4| .$ Find $g(3) .$ Find $g(x+4)$.

CC
Charles C.
Numerade Educator

Problem 37

Let $f(x)=x^{2}-6 x$. In each case, find all real numbers $x$ (if any) that satisfy the given equation.
(a) $f(x)=16$
(b) $f(x)=-10$
(c) $f(x)=-9$

CC
Charles C.
Numerade Educator

Problem 38

Let $g(t)=(4 t-6) /(t-4) .$ In each case, find all the real number solutions (if any) for the given equation.
(a) $g(t)=14$
(b) $g(t)=4$
(c) $g(t)=0$

CC
Charles C.
Numerade Educator

Problem 39

Refer to the demand function. In each case, solve the indicated equation for $n .$ Round each answer to the nearest integer and interpret the result.
$$p(n)=8$$

CC
Charles C.
Numerade Educator

Problem 40

Refer to the demand function. In each case, solve the indicated equation for $n .$ Round each answer to the nearest integer and interpret the result.
$$p(n)=18$$

CC
Charles C.
Numerade Educator

Problem 41

Refers to the demand function given in Example $7 .$ Now assume that a second economist proposes an alternative model f for the demand function: $$f(n)=4+\frac{3000}{n+100} \quad(100 \leq n \leq 1500)$$
where $n$ is the number of tee shirts that can be sold per month at a price of $f(n)$ dollars per shirt.
At a price level of $\$ 19$ per shirt, compare the predictions for monthly sales obtained using each model.

CC
Charles C.
Numerade Educator

Problem 42

Refers to the demand function given in Example $7 .$ Now assume that a second economist proposes an alternative model f for the demand function: $$f(n)=4+\frac{3000}{n+100} \quad(100 \leq n \leq 1500)$$
where $n$ is the number of tee shirts that can be sold per month at a price of $f(n)$ dollars per shirt.
Show that at a price level of $\$ 10$ per shirt, the model $p$ predicts more than twice as many sales per month as does the model $f .$

CC
Charles C.
Numerade Educator

Problem 43

Let $T(x)=2 x^{2}-3 x$. Find (and simplify) each expression.
(a) $T(x+2)$
(b) $T(x-2)$
(c) $T(x+2)-T(x-2)$

CC
Charles C.
Numerade Educator

Problem 44

Let $T(x)=2 x^{2}-3 x$. Find (and simplify) each expression.
(a) $T(x+h)$
(b) $T(x-h)$
(c) $T(x+h)-T(x-h)$

CC
Charles C.
Numerade Educator

Problem 45

Refer to the following table. The left-hand column of the table lists four errors to avoid in working with
function notation. In each case, use the function $f(x)=x^{2}-1$ and give a numerical example showing that the expressions on each side of the equation are not equal.
$$\begin{array}{|c|c|}\hline & \text { Numerical Example } \\& \text { Showing That the Equation } \\\text { Errors to Avoid } & \text { Is Not, in General, Valid } \\\hline f(a+b)=f(a)+f(b) & \\\hline\end{array}$$

CC
Charles C.
Numerade Educator

Problem 46

Refer to the following table. The left-hand column of the table lists four errors to avoid in working with
function notation. In each case, use the function $f(x)=x^{2}-1$ and give a numerical example showing that the expressions on each side of the equation are not equal.
$$\begin{array}{|c|c|} \hline \hline & \text { Numerical Example } \\& \text { Showing That the Equation } \\\text { Errors to Avoid } & \text { Is Not, in General, Valid } \\\hline f(a b)=f(a) \cdot f(b) &\end{array}$$

CC
Charles C.
Numerade Educator

Problem 47

Refer to the following table. The left-hand column of the table lists four errors to avoid in working with
function notation. In each case, use the function $f(x)=x^{2}-1$ and give a numerical example showing that the expressions on each side of the equation are not equal.
$$\begin{array}{|c|c|} \hline \hline & \text { Numerical Example } \\& \text { Showing That the Equation } \\\text { Errors to Avoid } & \text { Is Not, in General, Valid } \\\hline f\left(\frac{1}{a}\right)=\frac{1}{f(a)} &\end{array}$$

CC
Charles C.
Numerade Educator

Problem 48

Refer to the following table. The left-hand column of the table lists four errors to avoid in working with
function notation. In each case, use the function $f(x)=x^{2}-1$ and give a numerical example showing that the expressions on each side of the equation are not equal.
$$\begin{array}{|c|c|} \hline \hline & \text { Numerical Example } \\& \text { Showing That the Equation } \\\text { Errors to Avoid } & \text { Is Not, in General, Valid } \\\hline \frac{f(a)}{f(b)}=\frac{a}{b} &\end{array}$$

CC
Charles C.
Numerade Educator

Problem 49

Let $f(x)=(x-a) /(x+a)$.
(a) Find $f(a), f(2 a),$ and $f(3 a) .$ Is it true that $f(3 a)=f(a)+f(2 a) ?$
(b) Show that $f(5 a)=2 f(2 a)$.

CC
Charles C.
Numerade Educator

Problem 50

Let $k(x)=5 x^{3}+\frac{5}{x^{3}}-x-\frac{1}{x}$. Show that $k(x)=k(1 / x)$.

CC
Charles C.
Numerade Educator

Problem 51

Let $f(x)=2 x+3 .$ Find values for $a$ and $b$ such that the equation $f(a x+b)=x$ is true for all values of $x$.

CC
Charles C.
Numerade Educator

Problem 52

Let $f(t)=(t-x) /(t+y) .$ Show that
$$f(x+y)+f(x-y)=\frac{-2 y^{2}}{x^{2}+2 x y}$$

CC
Charles C.
Numerade Educator

Problem 53

Let $f(z)=\frac{3 z-4}{5 z-3} .$ Find $f\left(\frac{3 z-4}{5 z-3}\right)$.

CC
Charles C.
Numerade Educator

Problem 54

Let $F(x)=\frac{a x+b}{c x-a} .$ Show that $F\left(\frac{a x+b}{c x-a}\right)=x,$ where $a^{2}+b c \neq 0$.

CC
Charles C.
Numerade Educator

Problem 55

If $f(x)=-2 x^{2}+6 x+k$ and $f(0)=-1,$ find $k$.

CC
Charles C.
Numerade Educator

Problem 56

If $g(x)=x^{2}-3 x k-4$ and $g(1)=-2,$ find $k$.

CC
Charles C.
Numerade Educator

Problem 57

Let $h(x)=x^{2}-4 x-c .$ Find a nonzero value for $c$ such that $h(c)=c$.

CC
Charles C.
Numerade Educator

Problem 58

Let the function $L$ be defined by the following rule: $L(x)$ is the exponent to which 2 must be raised to yield $x$. (For the moment, we won't concern ourselves with the domain and range.) Then $L(8)=3,$ for example, since the exponent to which 2 must be raised to yield 8 is 3 (that is, $8=2^{3}$ ). Find the following outputs.
(a) $L(1)$
(b) $L(2)$
(c) $L(4)$
(d) $L(64)$
(e) $L(1 / 2)$
(f) $L(1 / 4)$
(g) $L(1 / 64)$
(h) $L(\sqrt{2})$
The function $L$ is called the logarithmic function with base $2 .$ The usual notation for $L(x)$ in this example is $\log _{2} x$ Logarithmic functions will be studied in Chapter 5 .

CC
Charles C.
Numerade Educator

Problem 59

Let $q(x)=a x^{2}+b x+c$. Evaluate
$$q\left(\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}\right)$$

CC
Charles C.
Numerade Educator

Problem 60

By definition, a fixed point for the function $f$ is a number $x_{0}$ such that $f\left(x_{0}\right)=x_{0} .$ For instance, to find any fixed points for the function $f(x)=3 x-2,$ we write $3 x_{0}-2=x_{0} .$ On solving this last equation, we find that $x_{0}=1 .$ Thus, 1 is a fixed point for $f$. Calculate the fixed points (if any) for each function.
(a) $f(x)=6 x+10$
(b) $g(x)=x^{2}-2 x-4$
(c) $S(t)=t^{2}$
(d) $R(z)=(z+1) /(z-1)$

CC
Charles C.
Numerade Educator

Problem 61

Consider the following two rules, $F$ and $G,$ where $F$ is the rule that assigns to each person his or her birth-mother and $G$ is the rule that assigns to each person his or her aunt. Explain why $F$ is a function but $G$ is not.

CC
Charles C.
Numerade Educator

Problem 62

Use this definition: A prime number is a positive whole number with no factors other than itself and 1. For example, $2,13,$ and 37 are primes, but 24 and 39 are not. By convention 1 is not considered prime, so the list of the first few primes is as follows:
$$2,3,5,7,11,13,17,19,23,29, \dots$$
Let $G$ be the rule that assigns to each positive integer the nearest prime. For example, $G(8)=7,$ since 7 is the prime nearest $8 .$ Explain why $G$ is not a function. How could you alter the definition of $G$ to make it a function? Note: There is more than one way to do this.

CC
Charles C.
Numerade Educator

Problem 63

Use this definition: A prime number is a positive whole number with no factors other than itself and 1. For example, $2,13,$ and 37 are primes, but 24 and 39 are not. By convention 1 is not considered prime, so the list of the first few primes is as follows:
Let $f$ be the function that assigns to each natural number $X$ the number of primes that are less than or equal to $x$. For example, $f(12)=5$ because, as you can easily check, five primes are less than or equal to $12 .$ Similarly, $f(3)=2$ because two primes are less than or equal to $3 .$ Find $f(8)$ $f(10),$ and $f(50)$.

CC
Charles C.
Numerade Educator

Problem 64

Use this definition: A prime number is a positive whole number with no factors other than itself and 1. For example, $2,13,$ and 37 are primes, but 24 and 39 are not. By convention 1 is not considered prime, so the list of the first few primes is as follows:
(a) If $P(x)=x^{2}-x+17,$ find $P(1), P(2), P(3),$ and $P(4)$ Can you find a natural number $x$ for which $P(x)$ is not prime?
(b) If $Q(x)=x^{2}-x+41,$ find $Q(1), Q(2), Q(3),$ and $Q(4)$ Can you find a natural number $x$ for which $Q(x)$ is not prime?

CC
Charles C.
Numerade Educator

Problem 65

$\pi=3.141592653589793 \ldots$ and so on!
For each natural number $n,$ let $G(n)$ be the digit in the $n$ th decimal place of $\pi .$ For instance, according to the expression for $\pi$ given above, we have $G(1)=1, G(2)=4,$ and $G(5)=9$.
(a) Use the expression for $\pi$ given above to evaluate $G(10)$ and $G(14)$
(b) Use the Internet to help you evaluate $G(100), G(750)$ and $G(1000) .$

CC
Charles C.
Numerade Educator

Problem 66

If $f(x)=m x+b,$ show that $\frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2}=f\left(\frac{x_{1}+x_{2}}{2}\right)$

CC
Charles C.
Numerade Educator

Problem 67

Let $f(x)=a x^{2}+b x+c,$ where $a<0 .$ Show that $\frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2} \leq f\left(\frac{x_{1}+x_{2}}{2}\right)$

CC
Charles C.
Numerade Educator

Problem 68

Find the range of the function defined by $y=\frac{x}{x^{2}+1}$.

CC
Charles C.
Numerade Educator