Problem 2
If
$$f(x)=\frac{x^{2}-x}{x-1} \quad \text { and } \quad g(x)=x$$
is it true that $f=g ?$
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?
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 $g .$
Problem 5
Figure 1 was recorded by an instrument operated by the Cali- fornia Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake.
Problem 6
In this section we discussed examples of ordinary, everyday functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.
Problem 7
$7-10$ Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.
Problem 8
$7-10$ Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.
Problem 9
$7-10$ Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.
Problem 10
$7-10$ Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.
Problem 11
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?
Problem 12
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.
Problem 13
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.
Problem 14
Three runners compete in a 100 -meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race?
Problem 15
The graph shows the power consumption for a day in September in San Francisco. (P is measured in megawatts; $t$ is measured in hours starting at midnight.)
(a) What was the power consumption at 6 AM? At 6 PM?
(b) When was the power consumption the lowest? When was it the highest? Do these times seem reasonable?
Problem 16
Sketch a rough graph of the number of hours of daylight as a function of the time of year.
Problem 17
Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.
Problem 18
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.
Problem 19
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.
Problem 20
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.
Problem 21
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.
Problem 22
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 building, 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.
Problem 23
The number $N$ (in millions) of US cellular phone subscribers is shown in the table. (Midyear estimates are given.)
$$\begin{array}{|c|c|c|c|c|c|}\hline t & {1996} & {1998} & {2000} & {2002} & {2004} & {2006} \\ \hline N & {44} & {69} & {109} & {141} & {182} & {233} \\ \hline\end{array}$$
(a) Use the data to sketch a rough graph of $N$ as a function of $t$
(b) Use your graph to estimate the number of cell-phone subscribers at midyear in 2001 and $2005 .$
Problem 24
Temperature readings $T$ (In$^{\circ} \mathrm{F}$) were recorded every two hours from midnight to $2 : 00 \mathrm{PM}$ in Phoenix on September $10,2008$ The time $t$ was measured in hours from midnight.
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline t & {0} & {2} & {4} & {6} & {8} & {10} & {12} & {14} \\ \hline T & {82} & {75} & {74} & {75} & {84} & {90} & {93} & {94} \\ \hline\end{array}$$
(a) Use the readings to sketch a rough graph of $T$ as a function of $t$ .
(b) Use your graph to estimate the temperature at $9 : 00$ AM.
Problem 25
If $f(x)=3 x^{2}-x+2$ find $f(2), \quad f(-2), f(a), f(-a)$ $f(a+1), 2 f(a), f(2 a), f\left(a^{2}\right),[f(a)]^{2}$ and $f(a+h)$
Problem 26
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.
Problem 27
$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=4+3 x-x^{2}, \quad \frac{f(3+h)-f(3)}{h}$$
Problem 28
$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=x^{3}, \quad \frac{f(a+h)-f(a)}{h}$$
Problem 29
$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=\frac{1}{x}, \quad \frac{f(x)-f(a)}{x-a}$$
Problem 30
$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=\frac{x+3}{x+1}, \quad \frac{f(x)-f(1)}{x-1}$$
Problem 32
$31-37$ Find the domain of the function.
$$f(x)=\frac{2 x^{3}-5}{x^{2}+x-6}$$
Problem 35
$31-37$ Find the domain of the function.
$$h(x)=\frac{1}{\sqrt[4]{x^{2}-5 x}}$$
Problem 36
$31-37$ Find the domain of the function.
$$f(u)=\frac{u+1}{1+\frac{1}{u+1}}$$
Problem 38
Find the domain and range and sketch the graph of the function $h(x)=\sqrt{4-x^{2}}$
Problem 39
$39-50$ Find the domain and sketch the graph of the function.
$$f(x)=2-0.4 x$$
Problem 40
$39-50$ Find the domain and sketch the graph of the function.
$$F(x)=x^{2}-2 x+1$$
Problem 41
$39-50$ Find the domain and sketch the graph of the function.
$$f(t)=2 t+t^{2}$$
Problem 42
$39-50$ Find the domain and sketch the graph of the function.
$$H(t)=\frac{4-t^{2}}{2-t}$$
Problem 43
$39-50$ Find the domain and sketch the graph of the function.
$$g(x)=\sqrt{x-5}$$
Problem 44
$39-50$ Find the domain and sketch the graph of the function.
$$F(x)=|2 x+1|$$
Problem 45
$39-50$ Find the domain and sketch the graph of the function.
$$G(x)=\frac{3 x+|x|}{x}$$
Problem 46
$39-50$ Find the domain and sketch the graph of the function.
$$g(x)=|x|-x$$
Problem 47
$39-50$ Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x<0} \\ {1-x} & {\text { if } x \geqslant 0}\end{array}\right.$$
Problem 48
$39-50$ Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{3-\frac{1}{2} x} & {\text { if } x \leqslant 2} \\ {2 x-5} & {\text { if } x>2}\end{array}\right.$$
Problem 49
$39-50$ Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x \leqslant-1} \\ {x^{2}} & {\text { if } x>-1}\end{array}\right.$$
Problem 50
$39-50$ Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{x+9} & {\text { if } x<-3} \\ {-2 x} & {\text { if }|x| \leqslant 3} \\ {-6} & {\text { if } x>3}\end{array}\right.$$
Problem 51
\begin{equation}
\begin{array}{l}{51-56 \text { Find an expression for the function whose graph is the }} \\ {\text { given curve. }}\end{array}
\end{equation}
The line segment joining the points $(1,-3)$ and $(5,7)$
Problem 52
\begin{equation}
\begin{array}{l}{51-56 \text { Find an expression for the function whose graph is the }} \\ {\text { given curve. }}\end{array}
\end{equation}
The line segment joining the points $(-5,10)$ and $(7,-10)$
Problem 53
\begin{equation}
\begin{array}{l}{51-56 \text { Find an expression for the function whose graph is the }} \\ {\text { given curve. }}\end{array}
\end{equation}
The bottom half of the parabola $x+(y-1)^{2}=0$
Problem 54
\begin{equation}
\begin{array}{l}{51-56 \text { Find an expression for the function whose graph is the }} \\ {\text { given curve. }}\end{array}
\end{equation}
The top half of the circle $x^{2}+(y-2)^{2}=4$
Problem 55
$51-56$ Find an expression for the function whose graph is the given curve.
Problem 56
$51-56$ Find an expression for the function whose graph is the given curve.
Problem 57
\begin{equation}
\begin{array}{l}{57-61 \text { Find a formula for the described function and state its }} \\ {\text { domain. }}\end{array}
\end{equation}
A rectangle has perimeter 20 $\mathrm{m} .$ Express the area of the rect- angle as a function of the length of one of its sides.
Problem 58
\begin{equation}
\begin{array}{l}{57-61 \text { Find a formula for the described function and state its }} \\ {\text { domain. }}\end{array}
\end{equation}
A rectangle has area 16 $\mathrm{m}^{2} .$ Express the perimeter of the rectangle as a function of the length of one of its sides.
Problem 59
\begin{equation}
\begin{array}{l}{57-61 \text { Find a formula for the described function and state its }} \\ {\text { domain. }}\end{array}
\end{equation}
Express the area of an equilateral triangle as a function of the length of a side.
Problem 60
\begin{equation}
\begin{array}{l}{57-61 \text { Find a formula for the described function and state its }} \\ {\text { domain. }}\end{array}
\end{equation}
Express the surface area of a cube as a function of its volume.
Problem 61
\begin{equation}
\begin{array}{l}{57-61 \text { Find a formula for the described function and state its }} \\ {\text { domain. }}\end{array}
\end{equation}
An open rectangular box with volume 2 $\mathrm{m}^{3}$ has a square base. Express the surface area of the box as a function of the length of a side of the base.
Problem 62
A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 $\mathrm{ft}$ , express the area $A$ of the window as a function of the width $x$ of the window.
Problem 63
A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side $x$ at each corner and then folding up the sides as in the figure. Express the volume $V$ of the box as a function of $X .$
Problem 64
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$\leqslant x \leqslant 600 .$
Problem 65
In a certain state the maximum speed permitted on freeways is 65 mi/h and the minimum speed is 40 $\mathrm{mi} / \mathrm{h}$ . The fine for violating these limits is $\$ 15$ for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fine $F$ as a function of the driving speed $x$ and $\operatorname{graph} F(x)$ for 0$\leqslant x \leqslant 100.$
Problem 66
An electricity company charges its customers a base rate of $\$ 10$ a month, plus 6 cents per kilowatt-hour $(\mathrm{kWh})$ for the first 1200 $\mathrm{kWh}$ and 7 cents per kWh for all usage over 1200 $\mathrm{kWh}$. Express the monthly cost $E$ as a function of the amount $x$ of electricity used. Then graph the function $E$ for 0$\leqslant x \leqslant 2000.$
Problem 67
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.
Problem 68
The functions in Example 10 and Exercise 67 are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life.
Problem 69
$69-70$ Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.
Problem 70
$69-70$ Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.
Problem 71
(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?
Problem 72
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
Problem 73
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=\frac{x}{x^{2}+1}$$
Problem 74
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=\frac{x^{2}}{x^{4}+1}$$
Problem 75
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=\frac{x}{x+1}$$
Problem 76
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=x|x|$$
Problem 77
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=1+3 x^{2}-x^{4}$$
Problem 78
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
$$f(x)=1+3 x^{3}-x^{5}$$
Problem 79
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.
Problem 80
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.
