Section 1
Four Ways to Represent a Function
If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it truethat $f=g ?$
If $$f(x)=\frac{x^{2}-x}{x-1} \quad$$ and $$\quad g(x)=x$$ is it true that $f=g ?$
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?
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 $g$ .
Figure 1 was recorded by an instrument operated by theCalifornia Department of Mines and Geology at the UniversityHospital of the University of Southern California in LosAngeles. Use it to estimate the range of the vertical groundacceleration function at USC during the Northridge earthquake.
In this section we discussed examples of ordinary, everydayfunctions: Population is a function of time, postage cost is afunction of weight, water temperature is a function of time. Givethree other examples of functions from everyday life that aredescribed verbally. What can you say about the domain andrange of each of your functions? If possible, sketch a roughgraph of each function.
$7-10$ Determine whether the curve is the graph of a function of $x .$If it is, state the domain and range of the function.
Shown is a graph of the global average temperature $T$ duringthe 20 th century. Estimate the following.(a) The global average temperature in 1950(b) The year when the average temperature was $14.2^{\circ} \mathrm{C}$(c) The year when the temperature was smallest; the year itwas largest(d) The range of $T$
Trees grow faster and form wider rings in warm years and grow more slowly and form narrower rings in cooler years. The figure shows ring widths of a Siberian pine from 1500 to $2000 .$(a) What is the range of the ring width function?(b) What does the graph tend to say about the temperatureof the earth? Does the graph reflect the volcanic erup-tions of the mid-19th century?
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 roughgraph of the temperature of the water as a function of the elapsed time.
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?
The graph shows the power consumption for a day in Septem-ber in San Francisco. $(P$ is measured in megawatts; $t$ is mea-sured in hours starting at midnight.)(a) What was the power consumption at 6 $\mathrm{AM} ?$ At 6 $\mathrm{PM}$ ?(b) When was the power consumption at 6 $\mathrm{AM}$ ? At 6 $\mathrm{PM}$ ?it the highest? Do these times seem reasonable?
Sketch a rough graph of the number of hours of daylight as a function of the time of year.
Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.
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.
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.
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.
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.
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.
Temperature readings T (in $^{\circ} \mathrm{F}$) were recorded every two hoursfrom midnight to $2 : 00 \mathrm{PM}$ in Atlanta on June $4,2013 .$ The time$t$ was measured in hours from midnight.$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline t & {0} & {2} & {4} & {6} & {8} & {10} & {12} & {14} \\ \hline T & {74} & {69} & {68} & {66} & {70} & {78} & {82} & {86} \\ \hline\end{array}$$(a) Use the readings to sketch a rough graph of $T$ as a functionof $t .$(b) Use your graph to estimate the temperature at $9 : 00$ AM.
Researchers measured the blood alcohol concentration $(\mathrm{BAC})$ of eight adult male subjects after rapid consumption of 30 $\mathrm{mL}$ of ethanol (corresponding to two standard alcoholic drinks). The table shows the data they obtained by averaging the BAC(in $\mathrm{g} / \mathrm{dL}$ ) of the eight men.(a) Use the readings to sketch the graph of the BAC as afunction of $t .$(b) Use your graph to describe how the effect of alcoholvaries with time.
\begin{equation}\begin{array}{l}{\text { If } f(x)=3 x^{2}-x+2, \text { 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}, \text { and } f(a+h)}\end{array}\end{equation}
A spherical balloon with radius $r$ inches has volume$V(r)=\frac{4}{3} \pi r^{3}$ . Find a function that represents the amount ofair required to inflate the balloon from a radius of $r$ inchesto a radius of $r+1$ inches.
$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.$$f(x)=4+3 x-x^{2}, \quad \frac{f(3+h)-f(3)}{h}$$
$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.$$f(x)=x^{3}, \quad \frac{f(a+h)-f(a)}{h}$$
$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.$$f(x)=\frac{1}{x}, \quad \frac{f(x)-f(a)}{x-a}$$
$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.$$f(x)=\frac{x+3}{x+1}, \quad \frac{f(x)-f(1)}{x-1}$$
$31-37$ Find the domain of the function.$$f(x)=\frac{x+4}{x^{2}-9}$$
$31-37$ Find the domain of the function.$$f(x)=\frac{2 x^{3}-5}{x^{2}+x-6}$$
$31-37$ Find the domain of the function.$$f(t)=\sqrt[3]{2 t-1}$$
$31-37$ Find the domain of the function.$$g(t)=\sqrt{3-t}-\sqrt{2+t}$$
$31-37$ Find the domain of the function.$$h(x)=\frac{1}{\sqrt[4]{x^{2}-5 x}}$$
$31-37$ Find the domain of the function.$$f(u)=\frac{u+1}{1+\frac{1}{u+1}}$$
$31-37$ Find the domain of the function.$$F(p)=\sqrt{2-\sqrt{p}}$$
38. Find the domain and range and sketch the graph of thefunction $h(x)=\sqrt{4-x^{2}}$
$39-40$ Find the domain and sketch the graph of the function.$$f(x)=1.6 x-2.4$$
$39-40$ Find the domain and sketch the graph of the function.$$g(t)=\frac{t^{2}-1}{t+1}$$
$41-44$ Evaluate $f(-3), f(0),$ and $f(2)$ for the piecewise definedfunction. Then sketch the graph of the function.$$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x<0} \\ {1-x} & {\text { if } x \geqslant 0}\end{array}\right.$$
$41-44$ Evaluate $f(-3), f(0),$ and $f(2)$ for the piecewise definedfunction. Then sketch the graph of the function.$$f(x)=\left\{\begin{array}{ll}{3-\frac{1}{2} x} & {\text { if } x<2} \\ {2 x-5} & {\text { if } x \geqslant 2}\end{array}\right.$$
$41-44$ Evaluate $f(-3), f(0),$ and $f(2)$ for the piecewise definedfunction. Then sketch the graph of the function.$$f(x)=\left\{\begin{array}{ll}{x+1} & {\text { if } x \leqslant-1} \\ {x^{2}} & {\text { if } x>-1}\end{array}\right.$$
$41-44$ Evaluate $f(-3), f(0),$ and $f(2)$ for the piecewise definedfunction. Then sketch the graph of the function.$$f(x)=\left\{\begin{array}{ll}{-1} & {\text { if } x \leqslant 1} \\ {7-2 x} & {\text { if } x>1}\end{array}\right.$$
$45-50$ Sketch the graph of the function.$$f(x)=x+|x|$$
$45-50$ Sketch the graph of the function.$$f(x)=|x+2|$$
$45-50$ Sketch the graph of the function.$$g(t)=|1-3 t|$$
$45-50$ Sketch the graph of the function.$$h(t)=|t|+|t+1|$$
$45-50$ Sketch the graph of the function.$$f(x)=\left\{\begin{array}{ll}{|x|} & {\text { if }|x| \leqslant 1} \\ {1} & {\text { if }|x|>1}\end{array}\right.$$
$45-50$ Sketch the graph of the function.$$g(x)=\| x|-1|$$
$51-56$ Find an expression for the function whose graph is the given curve.The line segment joining the points $(1,-3)$ and $(5,7)$
$51-56$ Find an expression for the function whose graph is the given curve.The line segment joining the points $(-5,10)$ and $(7,-10)$
$51-56$ Find an expression for the function whose graph is the given curve.The bottom half of the parabola $x+(y-1)^{2}=0$
$51-56$ Find an expression for the function whose graph is the given curve.The top half of the circle $x^{2}+(y-2)^{2}=4$
$51-56$ Find an expression for the function whose graph is the given curve.
$57-61$ Find a formula for the described function and state its domain.A rectangle has perimeter 20 $\mathrm{m} .$ Express the area of therectangle as a function of the length of one of its sides.
$57-61$ Find a formula for the described function and state its domain.
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.
Express the area of an equilateral triangle as a function of thelength of a side.
$57-61$ Find a formula for the described function and state its domain.A closed rectangular box with volume 8 $\mathrm{ft}^{3}$ has length twice thewidth. Express the height of the box as a function of the width.
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 lengthof a side of the base.
A Norman window has the shape of a rectangle surmountedby a semicircle. If the perimeter of the window is 30 ft,express the area A of the window as a function of the widthx of the window.
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.
A cell phone plan has a basic charge of $\$ 35$ a month. Theplan includes 400 free minutes and charges 10 cents for eachadditional minute of usage. Write the monthly cost $C$ as afunction of the number $x$ of minutes used and graph $C$ as afunction of $x$ for $0 \leq x \leqslant 600 .$
In a certain state the maximum speed permitted on freewaysis 65 $\mathrm{mi} / \mathrm{h}$ and the minimum speed is 40 $\mathrm{mi} / \mathrm{h}$ . The fine for violating these limits is $\$ 15$ for every mile per hour above themaximum speed or below the minimum speed. Express theamount of the fine $F$ as a function of the driving speed $x$ andgraph $F(x)$ for 0$\leqslant x \leqslant 100 .$
An electricity company charges its customers a base rateof $\$ 10$ a month, plus 6 cents per kilowatt-hour $(\mathrm{kWh})$ forthe first 1200 $\mathrm{kWh}$ and 7 cents per kWh for all usage over1200 $\mathrm{kWh} .$ Express the monthly cost $E$ as a function of the amount $x$ of electricity used. Then graph the function $E$ for0$\leqslant x \leqslant 2000$
In a certain country, income tax is assessed as follows. Thereis 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 $\$ 20,000 .$ Anyincome over $\$ 20,000$ is taxed at 15$\%$ .(a) Sketch the graph of the tax rate $R$ as a function of theincome $I .$(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 functionof the income I.
The functions in Example 10 and Exercise 67 are called stepfunctions because their graphs look like stairs. Give two otherexamples of step functions that arise in everyday life.
$69-70$ Graphs of $f$ and $g$ are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.
(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, whatother point must also be on the graph?
A function $f$ has domain $[-5,5]$ and a portion of its graphis 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.
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$$f(x)=\frac{x}{x^{2}+1}$$
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$$f(x)=\frac{x^{2}}{x^{4}+1}$$
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$$f(x)=\frac{x}{x+1}$$
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$$f(x)=x|x|$$
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$$f(x)=1+3 x^{2}-x^{4}$$
$73-78$ Determine whether $f$ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.$$f(x)=1+3 x^{3}-x^{5}$$
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 $g$ isodd? Justify your answers.
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.