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Chapter 1

FUNCTIONS AND LIMITS

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Problem 1

$$\begin{array}{l}{\text { If } f(x)=x+\sqrt{2-x} \text { and } g(u)=u+\sqrt{2-u}, \text { is it true }} \\ {\text { that } f=g^{\prime}}\end{array}$$

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Problem 2

If $$f(x)=\frac{x^{2}-x}{x-1} \quad \text { and } \quad g(x)=x$$
is it true that f=g?

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Problem 3

The graph of a function $f$ is given.
(a) State the value of $f(1) .$
(b) Estimate the value of $f(-1)$
(c) For what values of $x$ is $f(x)=1 ?$
(d) Estimate the value of $x$ such that $f(x)=0$ .
(e) State the domain and range of $f$ .
(f) On what interval is $f$ increasing?

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Problem 4

The graphs of $f$ and $g$ are given.
(a) State the values of $f(-4)$ and $g(3) .$
(b) For what values of $x$ is $f(x)=g(x) ?$
(c) Estimate the solution of the equation $f(x)=-1$
(d) On what interval is $f$ decreasing?
(e) State the domain and range of $f .$
(f) State the domain and range of $g .$

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Problem 5

Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.

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Problem 6

Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.

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Problem 7

Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.

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Problem 8

Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.

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Problem 9

The graph shown gives the weight of a certain person as a function of age. Describe in words how this person"s weight varies over time. What do you think happened when this person was 30 years old?

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Problem 10

The graph shows the height of the water in a bathtub as a function of time. Give a verbal description of what you think happened.

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Problem 11

You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.

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Problem 12

Sketch a rough graph of the number of hours of daylight as a function of the time of year.

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Problem 13

Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.

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Problem 14

Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.

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Problem 15

Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.

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Problem 16

You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.

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Problem 17

A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.

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Problem 18

An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If $t$ represents the time in minutes since the plane has left the terminal, let $x(t)$ be the horizontal distance traveled and $y(t)$ be the altitude of the plane.
(a) Sketch a possible graph of $x(t) .$
(b) Sketch a possible graph of $y(t) .$
(c) Sketch a possible graph of the ground speed.
(d) Sketch a possible graph of the vertical velocity.

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Problem 19

If $f(x)=3 x^{2}-x+2,$ find $f(2), f(-2), f(a), f(-a)$ $f(a+1), 2 f(a), f(2 a), f\left(a^{2}\right),[f(a)]^{2},$ and $f(a+h)$

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Problem 20

A spherical balloon with radius $r$ inches has volume $V(r)=\frac{4}{3} \pi r^{3} .$ Find a function that represents the amount of air required to inflate the balloon from a radius of $r$ inches to a radius of $r+1$ inches.

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Problem 21

Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=4+3 x-x^{2}, \quad \frac{f(3+h)-f(3)}{h}$$

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Problem 22

Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=x^{3}, \quad \frac{f(a+h)-f(a)}{h}$$

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Problem 23

Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=\frac{1}{x}, \quad \frac{f(x)-f(a)}{x-a}$$

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Problem 24

Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=\frac{x+3}{x+1}, \quad \frac{f(x)-f(1)}{x-1}$$

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Problem 25

Find the domain of the function.
$$f(x)=\frac{x+4}{x^{2}-9}$$

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Problem 26

Find the domain of the function.
$$f(x)=\frac{2 x^{3}-5}{x^{2}+x-6}$$

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Problem 27

Find the domain of the function.
$$F(p)=\sqrt{2-\sqrt{p}}$$

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Problem 28

Find the domain of the function.
$$g(t)=\sqrt{3-t}-\sqrt{2+t}$$

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Problem 29

Find the domain of the function.
$$h(x)=\frac{1}{\sqrt[4]{x^{2}-5 x}}$$

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Problem 30

Find the domain and range and sketch the graph of the function $h(x)=\sqrt{4-x^{2}}$

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Problem 31

Find the domain and sketch the graph of the function.
$$f(x)=2-0.4 x$$

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Problem 32

Find the domain and sketch the graph of the function.
$$F(x)=x^{2}-2 x+1$$

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Problem 33

Find the domain and sketch the graph of the function.
$$f(t)=2 t+t^{2}$$

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Problem 34

Find the domain and sketch the graph of the function.
$$H(t)=\frac{4-t^{2}}{2-t}$$

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Problem 35

Find the domain and sketch the graph of the function.
$$g(x)=\sqrt{x-5}$$

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Problem 36

Find the domain and sketch the graph of the function.
$$F(x)=|2 x+1|$$

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Problem 37

Find the domain and sketch the graph of the function.
$$G(x)=\frac{3 x+|x|}{x}$$

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Problem 38

Find the domain and sketch the graph of the function.
$$g(x)=|x|-x$$

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Problem 39

Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x<0} \\ {1-x} & {\text { if } x \geqq 0}\end{array}\right.$$

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Problem 40

Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{3-\frac{1}{2} x} & {\text { if } x \leqslant 2} \\ {2 x-5} & {\text { if } x>2}\end{array}\right.$$

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Problem 41

Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x \leqslant-1} \\ {x^{2}} & {\text { if } x>-1}\end{array}\right.$$

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Problem 42

Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{-1} & {\text { if } x \leqslant-1} \\ {3 x+2} & {\text { if }|x|<1} \\ {7-2 x} & {\text { if } x \geqslant 1}\end{array}\right.$$

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Problem 43

Find an expression for the function whose graph is the given curve.
The line segment joining the points $(1,-3)$ and $(5,7)$

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Problem 44

Find an expression for the function whose graph is the given curve.
The line segment joining the points $(-5,10)$ and $(7,-10)$

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Problem 45

Find an expression for the function whose graph is the given curve.
The bottom half of the parabola $x+(y-1)^{2}=0$

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Problem 46

Find an expression for the function whose graph is the given curve.
The top half of the circle $x^{2}+(y-2)^{2}=4$

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Problem 47

Find a formula for the described function and state its domain.
$$\begin{array}{l}{\text { A rectangle has perimeter } 20 \mathrm{m} \text { . Fxpress the area of the }} \\ {\text { rectangle as a function of the length of one of its sides. }}\end{array}$$

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Problem 48

Find a formula for the described function and state its domain.
$$\begin{array}{l}{\text { A rectangle has area } 16 \mathrm{m}^{2} \text { . Express the perimeter of the }} \\ {\text { rectangle as a function of the length of one of its sides. }}\end{array}$$

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Problem 49

Find a formula for the described function and state its domain.
$$\begin{array}{l}{\text { Express the area of an equilateral triangle as a function of }} \\ {\text { the length of a side. }}\end{array}$$

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Problem 50

Find a formula for the described function and state its domain.
$$\begin{array}{l}{\text { Express the surface area of a cube as a function of its }} \\ {\text { volume. }}\end{array}$$

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Problem 51

Find a formula for the described function and state its domain.
$$\begin{array}{l}{\text { An open rectangular box with volume } 2 \mathrm{m}^{3} \text { has a square }} \\ {\text { base. Express the surface area of the box as a function of }} \\ {\text { the length of a side of the base. }}\end{array}$$

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Problem 52

A cell phone plan has a basic charge of $\$ 35$ a month. The
plan includes 400 free minutes and charges 10 cents for
each additional minute of usage. Write the monthly cost $C$
as a function of the number $x$ of minutes used and graph $C$
as a function of $x$ for 0$\leqslant x \leqslant 600 .$

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Problem 53

In a certain country, income tax is assessed as follows.
There is no tax on income up to $\$ 10,000 .$ Any income over
$\$ 10,000$ is taxed at a rate of $10 \%,$ up to an income of
$\$ 20,000$ . Any income over $\$ 20,000$ is taxed at 15$\%$ .
(a) Sketch the graph of the tax rate $R$ as a function of the
income $I .$
(b) How much tax is assessed on an income of $\$ 14,000 ?$
On $\$ 26,000 ?$
(c) Sketch the graph of the total assessed tax $T$ as a function
of the income $I .$

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Problem 54

The functions in Example 6 and Exercises 52 and 53$(\mathrm{a})$ are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life.

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Problem 55

Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.

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Problem 56

Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.

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Problem 57

(a) If the point $(5,3)$ is on the graph of an even function, what other point must also be on the graph?
(b) If the point $(5,3)$ is on the graph of an odd function, what other point must also be on the graph?

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Problem 58

A function $f$ has domain $[-5,5]$ and a portion of its graph is shown.
(a) Complete the graph of $f$ if it is known that $f$ is even.
(b) Complete the graph of $f$ if it is known that $f$ is odd.

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Problem 59

Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=\frac{x}{x^{2}+1}$$

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Problem 60

Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=\frac{x^{2}}{x^{4}+1}$$

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Problem 61

Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=\frac{x}{x+1}$$

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Problem 62

Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=x|x|$$

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Problem 63

Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=1+3 x^{2}-x^{4}$$

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Problem 64

Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=1+3 x^{3}-x^{5}$$

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Problem 65

If $f$ and $g$ are both even functions, is $f+g$ even? If $f$ and $g$
are both odd functions, is $f+g$ odd? What if $f$ is even and
$g$ is odd? Justify your answers.

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Problem 66

If $f$ and $g$ are both even functions, is the product $f g$ even $?$ If
$f$ and $g$ are both odd functions, is $f g$ odd? What if $f$ is even
and $g$ is odd? Justify your answers.

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