Calculus, Early Transcendentals

Dennis G. Zill, Warren S. Wright

Chapter 1

Functions - all with Video Answers

Educators

Section 1

Functions and Graphs

Problem 1

In Problems $1-6,$ find the indicated function values.
If $f(x)=x^{2}-1 ; \quad f(-5), f(-\sqrt{3}), f(3),$ and $f(6)$

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Problem 2

Find the indicated function values.
If $f(x)=-2 x^{2}+x ; \quad f(-5), f\left(-\frac{1}{2}\right), f(2),$ and $f(7)$

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Problem 3

Find the indicated function values.
If $f(x)=\sqrt{x+1} ; \quad f(-1), f(0), f(3),$ and $f(5)$

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Problem 4

Find the indicated function values.
If $f(x)=\sqrt{2 x+4} ; \quad f\left(-\frac{1}{2}\right), f\left(\frac{1}{2}\right), f\left(\frac{5}{2}\right),$ and

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Problem 5

Find the indicated function values.
If $f(x)=\frac{3 x}{x^{2}+1} ; \quad f(-1), f(0), f(1),$ and $f(\sqrt{2})$

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Problem 6

Find the indicated function values.
If $f(x)=\frac{x^{2}}{x^{3}-2} ; \quad f(-\sqrt{2}), f(-1), f(0),$ and $f\left(\frac{1}{2}\right)$

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Problem 7

In Problems 7 and $8,$ find $$
f(x), f(2 a), f\left(a^{2}\right), f(-5 x), f(2 a+1), f(x+h)
$$
for the given function $f$ and simplify as much as possible.
$f(\quad)=-2(\quad)^{2}+3(\quad)$

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Problem 8

Find $$
f(x), f(2 a), f\left(a^{2}\right), f(-5 x), f(2 a+1), f(x+h)
$$
for the given function $f$ and simplify as much as possible.
$f(\quad)=-2(\quad)^{2}+3(\quad)$
$f(\quad)=(\quad)^{3}-2(\quad)^{2}+20$

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Problem 9

For what values of $x$ is $f(x)=6 x^{2}-1$ equal to 23 ?

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Problem 10

For what values of $x$ is $f(x)=\sqrt{x-4}$ equal to $4 ?$

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Problem 11

In Problems $11-26,$ find the domain of the given function $f$.
$f(x)=\sqrt{4 x-2}$

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Problem 12

Find the domain of the given function $f$.
$f(x)=\sqrt{15-5 x}$

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Problem 13

Find the domain of the given function $f$.
$f(x)=\frac{10}{\sqrt{1-x}}$

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Problem 14

Find the domain of the given function $f$.
$f(x)=\frac{2 x}{\sqrt{3 x-1}}$

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Problem 15

Find the domain of the given function $f$.
$f(x)=\frac{2 x-5}{x(x-3)}$

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Problem 16

Find the domain of the given function $f$.
$f(x)=\frac{x}{x^{2}-1}$

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Problem 17

Find the domain of the given function $f$.
$f(x)=\frac{1}{x^{2}-10 x+25}$

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Problem 18

Find the domain of the given function $f$.
$f(x)=\frac{x+1}{x^{2}-4 x-12}$

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Problem 19

Find the domain of the given function $f$.
$f(x)=\frac{x}{x^{2}-x+1}$

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Problem 20

Find the domain of the given function $f$.
$f(x)=\frac{x^{2}-9}{x^{2}-2 x-1}$

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Problem 21

Find the domain of the given function $f$.
$f(x)=\sqrt{25-x^{2}}$

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Problem 22

Find the domain of the given function $f$.
$f(x)=\sqrt{x(4-x)}$

AG

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Problem 23

Find the domain of the given function $f$.
$f(x)=\sqrt{x^{2}-5 x}$

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Problem 24

Find the domain of the given function $f$.
$f(x)=\sqrt{x^{2}-3 x-10}$

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Problem 25

Find the domain of the given function $f$.
$f(x)=\sqrt{\frac{3-x}{x+2}}$

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Problem 26

Find the domain of the given function $f$.
$f(x)=\sqrt{\frac{5-x}{x}}$

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Problem 27

In Problems $27-30,$ determine whether the graph in the figure is the graph of a function.

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Problem 28

In Problems $27-30,$ determine whether the graph in the figure is the graph of a function.

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Problem 29

In Problems $27-30,$ determine whether the graph in the figure is the graph of a function.

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Problem 30

In Problems $27-30,$ determine whether the graph in the figure is the graph of a function.

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Problem 31

In Problems $31-34,$ use the graph of the function $f$ given in the fioure to find its domain and range

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Problem 32

In Problems $31-34,$ use the graph of the function $f$ given in the fioure to find its domain and range

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Problem 33

In Problems $31-34,$ use the graph of the function $f$ given in the fioure to find its domain and range

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Problem 34

In Problems $31-34,$ use the graph of the function $f$ given in the fioure to find its domain and range

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Problem 35

In Problems $35-44,$ find the $x$ - and $y$ -intercepts, if any, of the graph of the given function $f$. Do not graph.
$f(x)=\frac{1}{2} x-4$

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Problem 36

Find the $x$ - and $y$ -intercepts, if any, of the graph of the given function $f$. Do not graph.
$f(x)=x^{2}-6 x+5$

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Problem 37

Find the $x$ - and $y$ -intercepts, if any, of the graph of the given function $f$. Do not graph.
$f(x)=4(x-2)^{2}-1$

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Problem 38

Find the $x$ - and $y$ -intercepts, if any, of the graph of the given function $f$. Do not graph.
$f(x)=(2 x-3)\left(x^{2}+8 x+16\right)$

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Problem 39

Find the $x$ - and $y$ -intercepts, if any, of the graph of the given function $f$. Do not graph.
$f(x)=x^{3}-x^{2}-2 x$

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Problem 40

Find the $x$ - and $y$ -intercepts, if any, of the graph of the given function $f$. Do not graph.
$f(x)=x^{4}-1$

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Problem 41

Find the $x$ - and $y$ -intercepts, if any, of the graph of the given function $f$. Do not graph.
$f(x)=\frac{x^{2}+4}{x^{2}-16}$

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Problem 42

Find the $x$ - and $y$ -intercepts, if any, of the graph of the given function $f$. Do not graph.
$f(x)=\frac{x(x+1)(x-6)}{x+8}$

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Problem 43

Find the $x$ - and $y$ -intercepts, if any, of the graph of the given function $f$. Do not graph.
$f(x)=\frac{3}{2} \sqrt{4-x^{2}}$

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Problem 44

Find the $x$ - and $y$ -intercepts, if any, of the graph of the given function $f$. Do not graph.
$f(x)=\frac{1}{2} \sqrt{x^{2}-2 x-3}$

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Problem 45

In Problems 45 and $46,$ use the graph of the function $f$ given in the figure to estimate the values of $f(-3), f(-2), f(-1), f(1),$ $f(2),$ and $f(3) .$ Estimate the $y$ -intercept.

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Problem 46

In Problems 45 and $46,$ use the graph of the function $f$ given in the figure to estimate the values of $f(-3), f(-2), f(-1), f(1),$ $f(2),$ and $f(3) .$ Estimate the $y$ -intercept.

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Problem 47

In Problems 47 and $48,$ use the graph of the function $f$ given in the figure to estimate the values of $f(-2), f(-1.5), f(0.5)$ $f(1), f(2),$ and $f(3.2) .$ Estimate the $x$ -intercepts.

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Problem 48

In Problems 47 and $48,$ use the graph of the function $f$ given in the figure to estimate the values of $f(-2), f(-1.5), f(0.5)$ $f(1), f(2),$ and $f(3.2) .$ Estimate the $x$ -intercepts.

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Problem 49

In Problems 49 and $50,$ find two functions $y=f_{1}(x)$ and $y=f_{2}(x)$ defined by the given equation. Find the domain of the functions $f_{1}$ and $f_{2}$.
$x=y^{2}-5$

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Problem 50

Find two functions $y=f_{1}(x)$ and $y=f_{2}(x)$ defined by the given equation. Find the domain of the functions $f_{1}$ and $f_{2}$.
$x^{2}-4 y^{2}=16$

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Problem 51

Some of the functions that you will encounter later on in this text will have the set of positive integers $n$ as their domain. The factorial function $f(n)=n !$ is defined as the product of the first $n$ positive integers, that is,
$$
f(n)=n !=1 \cdot 2 \cdot 3 \cdots(n-1) \cdot n
$$
(a) Evaluate $f(2), f(3), f(5),$ and $f(7)$.
(b) Show that $f(n+1)=f(n) \cdot(n+1)$.
(c) Simplify $f(5) / f(4)$ and $f(7) / f(5)$.
(d) Simplify $f(n+3) / f(n)$.

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Problem 52

Another function of a positive integer $n$ gives the sum of the first $n$ squared positive integers:
$$
S(n)=\frac{1}{6} n(n+1)(2 n+1)
$$
(a) Find the value of the sum $1^{2}+2^{2}+\cdots+99^{2}+100^{2}$.
(b) Find $n$ such that $300<S(n)<400$. [Hint: Use a calculator.]

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Problem 53

Determine an equation of a function $y=f(x)$ whose domain is
(a) $[3, \infty)$
(b) $(3, \infty)$.

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Problem 54

Determine an equation of a function $y=f(x)$ whose range is
(a) $[3, \infty)$
(b) $(3, \infty)$.

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Problem 55

From the graph of $f(x)=-x^{2}+2 x+3$ given in FIGURE 1.1 .23 determine the range and domain of the function $g(x)=\sqrt{f(x)} .$ Explain your reasoning in one or two sentences.

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Problem 56

Let $P$ denote any point $(x, f(x))$ on the graph of a function $f$. Suppose that the line segments $P T$ and $P S$ are perpendicular to the $x$ - and $y$ -axes, respectively. Let $M_{1}, M_{2}$, and $M_{3}$ be, in turn, the midpoints of $P T, P S,$ and $S T$ as shown in FIGURE 1.1.24. Find a function that describes the path of the points $M_{1} .$ Repeat for the midpoints $M_{2}$ and $M_{3}$.

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Problem 57

On page 7 we saw that the ceiling function $g(x)=\lceil x\rceil$ is defined to be the least integer $n$ that is greater than or equal to $x$. Fill in the blanks.

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Problem 58

Graph the ceiling function $g(x)=\lceil x\rceil$ defined in Problem 57 .

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Problem 59

The piecewise-defined function
$$
\operatorname{int}(x)=\left\{\begin{array}{ll}
\lfloor x\rfloor, & x \geq 0 \\
\lceil x\rceil, & x<0
\end{array}\right.
$$
is called the integer function. Graph $\operatorname{int}(x)$.

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Problem 60

Discuss how to graph the function $f(x)=|x|+|x-3|$. Carry out your ideas.

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Problem 61

In Problems 61 and $62,$ describe in words how the graphs of the given functions differ.
$f(x)=\frac{x^{2}-9}{x-3}$
$g(x)=\left\{\begin{array}{ll}\frac{x^{2}-9}{x-3}, & x \neq 3 \\ 4, & x=3\end{array}, \quad h(x)=\left\{\begin{array}{ll}\frac{x^{2}-9}{x-3}, & x \neq 3 \\ 6, & x=3\end{array}\right.\right.$

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Problem 62

Describe in words how the graphs of the given functions differ.
$f(x)=\frac{x^{4}-1}{x^{2}-1}$
$g(x)=\left\{\begin{array}{ll}\frac{x^{4}-1}{x-1}, & x \neq 1 \\ 0, & x=1\end{array}, \quad h(x)=\left\{\begin{array}{ll}\frac{x^{4}-1}{x^{2}-1}, & x \neq 1 \\ 2, & x=1\end{array}\right.\right.$

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