Chapter 3

The given function is not one-to-one. Restrict its domain so that the resulting function \(i\) s one-to-one. Find the inverse of the function with the restricted domain. There is more than one correct answer.) \(h(x)=(x+2)^{2}\)

Internet Purchases An Internet bookstore charges \(\$ 15\) shipping for orders under \(\$ 100\) , but provides free shipping for orders of \(\$ 100\) or more. The cost \(C\) of an order is a function of the total price \(x\) of the books purchased, given by $$ C(x)=\left\\{\begin{array}{ll}{x+15} & {\text { if } x<100} \\ {x} & {\text { if } x \geq 100}\end{array}\right. $$ (a) Find \(C(75), C(90), C(100),\) and \(C(105)\) (b) What do your answers in part (a) represent?

Determine whether the equation defines y as a function of x. (See Example 10.) $$ \sqrt{x}+y=12 $$

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = {x} +{\frac1x}$$

Volume of Water Between \(0^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C},\) the volume \(V\) (in cubic centimeters) of 1 \(\mathrm{kg}\) of water at a temperature \(T\) is given by the formula $$ V=999.87-0.06426 T+0.0085043 T^{2}-0.0000679 T^{3} $$ Find the temperature at which the volume of 1 \(\mathrm{kg}\) of water is a minimum.

The given function is not one-to-one. Restrict its domain so that the resulting function \(i\) s one-to-one. Find the inverse of the function with the restricted domain. There is more than one correct answer.) \(k(x)=|x-3|\)

Determine whether the equation defines y as a function of x. (See Example 10.) $$ 2|x|+y=0 $$

The graphs of \(f(x)=x^{2}-4\) and \(g(x)=\left|x^{2}-4\right|\) are shown. Explain how the graph of \(g\) is obtained from the graph of \(f .\)

Coughing When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. At the same time, the trachea contracts, causing the expelled air to move faster and increasing the pressure on the foreign object. According to a mathematical model coughing, the velocity \(v\) of the airstream through an average-sized person's trachea is related to the radius \(r\) of the trachea (in centimeters) by the function $$ v(r)=3.2(1-r) r^{2}, \quad \frac{1}{2} \leq r \leq 1 $$ Determine the value of \(r\) for which \(v\) is a maximum.

Determine whether the equation defines y as a function of x. (See Example 10.) $$ 2 x+|y|=0 $$

Maxima and Minima In Example 5 we saw a real-world situation in which the maximum value of a function is important. Name several other everyday situations in which a maximum or minimum value is important.

Use the graph of \(f\) to sketch the graph of \(f^{-1} .\)

Height of Grass A home owner 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 beginning on a Sunday.

Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose \(g(x)=\sqrt{f(x)},\) where \(f(x) \geq 0\) for all \(x\) . Explain why the local minima and maxima of \(f\) and \(g\) occur at the same values of \(x .\) (b) Let \(g(x)\) be the distance between the point \((3,0)\) and the point \(\left(x, x^{2}\right)\) on the graph of the parabola \(y=x^{2}\) Express \(g\) as a function of \(x .\) (c) Find the minimum value of the function \(g\) that you found in part (b). Use the principle described in part (a) to simplify your work.

For his services, a private investigator requires a \(\$ 500\) retention fee plus \(\$ 80\) per hour. Let \(x\) represent the number of hours the investigator spends working on a case. (a) Find a function \(f\) that models the investigator's fee as a function of \(x .\) (b) Find \(f^{-1} .\) What does \(f^{-1}\) represent? (c) Find \(f^{-1}(1220) .\) What does your answer represent?

Temperature Change 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. Sketch a rough graph of the temperature of the pie as a function of time.

Determine whether the equation defines y as a function of x. (See Example 10.) $$ x=y^{4} $$

71–72 ? Sketch the graph of each function. $$ f(x)=x^{3} \quad \text { (b) } g(x)=\left|x^{3}\right| $$

Maximum of a Fourth-Degree Polynomial Find the maximum value of the function $$ f(x)=3+4 x^{2}-x^{4} $$ \(\left[\text {Hint} : \text { Let } t=x^{2} .\right]\)

A tank holds 100 gallons of water, which drains from a leak at the bottom, causing the tank to empty in 40 minutes. Toricelli's Law gives the volume of water remaining in the tank after \(t\) minutes as $$V(t)=100\left(1-\frac{t}{40}\right)^{2}$$ (a) Find \(V^{-1} .\) What does \(V^{-1}\) represent? (b) Find \(V^{-1}(15) .\) What does your answer represent?

Daily Temperature Change Temperature readings \(T\) (in F) were recorded every 2 hours from midnight to noon in Atlanta, Georgia, on March \(18,1996 .\) The time \(t\) was measured in hours from midnight. Sketch a rough graph of \(T\) as a function of \(t\) \(\begin{array}{|c|c|}\hline t & {T} \\ \hline 0 & {58} \\ {2} & {57} \\ {4} & {53} \\ {6} & {50} \\ {8} & {51} \\ {10} & {57} \\ {12} & {61} \\\ \hline\end{array}\)

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=x^{2}+c$$ (a) \(c=0,2,4,6 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=0,-2,-4,-6 ;[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

Sales Growth The annual sales of a certain company can be modeled by the function \(f(t)=4+0.01 t^{2}\) , where \(t\) represents years since 1990 and \(f(t)\) is measured in millions of dollars. (a) What shifting and shrinking operations must be performed on the function \(y=t^{2}\) to obtain the function \(y=f(t) ?\) (b) Suppose you want \(t\) to represent years since 2000 instead of \(1990 .\) What transformation would you have to apply to the function \(y=f(t)\) to accomplish this? Write the new function \(y=g(t)\) that results from this transformation.

Population Growth The population \(P(\text { in thousands })\) of San Jose, California, from 1988 to 2000 is shown in the table. (Midyear estimates are given.) Draw a rough graph of \(P\) as a function of time \(t\). $$ \begin{array}{|c|c|}\hline t & {P} \\ \hline 1988 & {733} \\ {1990} & {782} \\\ {1992} & {800} \\ {1994} & {817} \\ {1996} & {838} \\ {1998} & {861} \\\ {2000} & {895} \\ \hline\end{array} $$

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=(x-c)^{2}$$ (a) \(c=0,1,2,3 ;[-5,5]\) by \([-10,10]\) (b) \(c=0,-1,-2,-3 ; \quad[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

Changing Temperature Scales The temperature on a certain afternoon is modeled by the function $$C(t)=\frac{1}{2} t^{2}+2$$ where \(t\) represents hours after 12 noon \((0 \leq t \leq 6),\) and \(C\) is measured in "C. (a) What shifting and shrinking operations must be performed on the function \(y=t^{2}\) to obtain the function \(y=C(t) ?\) (b) Suppose you want to measure the temperature in \(^{\circ} \mathrm{F}\) instead. What transformation would you have to apply to the function \(y=C(t)\) to accomplish this? (Use the fact that the relationship between Celsius and Fahrenheit degrees is given by \(F=\frac{9}{5} C+32 .\) ) Write the new function \(y=F(t)\) that results from this transformation.

The amount of a commodity sold is called the demand for the commodity. The demand \(D\) for a certain commodity is a function of the price given by $$D(p)=-3 p+150$$ (a) Find \(D^{-1} .\) What does \(D^{-1}\) represent? (b) Find \(D^{-1}(30 \text { ). What does your answer represent? }\)

Examples of Functions At the beginning of this section we discussed three examples of everyday, ordinary functions: Height is a function of age, temperature is a function of date, and postage cost is a function of weight. Give three other examples of functions from everyday life.

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=(x-c)^{3}$$ (a) \(c=0,2,4,6 ; \quad[-10,10]\) by \([-10,10]\) (b) \(c=0,-2,-4,-6 ; \quad[-10,10]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

Sums of Even and Odd Functions If \(f\) and \(g\) are both even functions, is \(f+g\) necessarily even? If both are odd, is their sum necessarily odd? What can you say about the sum if one is odd and one is even? In each case, prove your answer.

The relationship between the Fahrenheit \((F)\) and Celsius \((C)\) scales is given by $$F(C)=\frac{9}{5} C+32$$ (a) Find \(F^{-1} .\) What does \(F^{-1}\) represent? (b) Find \(F^{-1}(86) .\) What does your answer represent?

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=c x^{2}$$ (a) \(c=1, \frac{1}{2}, 2,4 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=1,-1,-\frac{1}{2},-2 ;[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

The relative value of currencies fluctuates every day. When this problem was written, one Canadian dollar was worth 0.8159 U.S. dollar. (a) Find a function \(f\) that gives the U.S. dollar value \(f(x)\) of \(x\) Canadian dollars. (b) Find \(f^{-1} .\) What does \(f^{-1}\) represent? (c) How much Canadian money would \(\$ 12,250\) in U.S. currency be worth?

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=x^{c}$$ (a) \(c=\frac{1}{2}, \frac{1}{4}, \frac{1}{6} ; \quad[-1,4]\) by \([-1,3]\) (b) \(c=1, \frac{1}{3}, \frac{1}{5} ; \quad[-3,3]\) by \([-2,2]\) (c) How does the value of \(c\) affect the graph?

Even and Odd Power Functions What must be true about the integer \(n\) if the function $$f(x)=x^{n}$$ is an even function? If it is an odd function? Why do you think the names "even" and "odd" were chosen for these function properties?

In a certain country, the tax on incomes less than or equal to \(€ 20,000\) is 10\(\% .\) For incomes more than \(€ 20,000,\) the tax is \(€ 2000\) plus 20\(\%\) of the amount over \(€ 20,000 .\) (a) Find a function \(f\) that gives the income tax on an income \(x\) . Express \(f\) as a piecewise defined function. (b) Find \(f^{-1} .\) What does \(f^{-1}\) represent? (c) How much income would require paying a tax of \(€ 10,000 ?\)

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=1 / x^{n}$$ (a) \(n=1,3 ; \quad[-3,3]\) by \([-3,3]\) (b) \(n=2,4 ; \quad[-3,3]\) by \([-3,3]\) (c) How does the value of \(n\) affect the graph?

A car dealership advertises a 15\(\%\) discount on all its new cars. In addition, the manufacturer offers a \(\$ 1000\) rebate on the purchase of a new car. Let \(x\) represent the sticker price of the car. (a) Suppose only the 15\(\%\) discount applies. Find a function \(f\) that models the purchase price of the car as a function of the sticker price \(x\) . (b) Suppose only the \(\$ 1000\) rebate applies. Find a function \(g\) that models the purchase price of the car as a function of the sticker price \(x .\) (c) Find a formula for \(H=f \circ g .\) (d) Find \(H^{-1}\) . What does \(H^{-1}\) represent? (e) Find \(H^{-1}(13,000) .\) What does your answer represent?

Find a function whose graph is the given curve. The line segment joining the points \((-2,1)\) and \((4,-6)\)

Marcello's Pizza charges a base price of \(\$ 7\) for a large pizza, plus \(\$ 2\) for each topping. Thus, if you order a large pizza with \(x\) toppings, the price of your pizza is given by the function \(f(x)=7+2 x .\) Find \(f^{-1} .\) What does the function \(f^{-1}\) represent?

Find a function whose graph is the given curve. The line segment joining the points \((-3,-2)\) and \((6,3)\)

For the linear function \(f(x)=m x+b\) to be one-to-one, what must be true about its slope? If it is one-to-one, find its inverse. Is the inverse linear? If so, what is its slope?

Find a function whose graph is the given curve. The top half of the circle \(x^{2}+y^{2}=9\)

In the margin notes in this section we pointed out that the inverse of a function can be found by simply reversing the operations that make up the function. For instance, in Example 6 we saw that the inverse of $$f(x)=3 x-2 \quad \text { is } \quad f^{-1}(x)=\frac{x+2}{3}$$ because the "reverse" of "multiply by 3 and subtract 2 " is "add 2 and divide by 3 " Use the same procedure to find the inverse of the following functions. (a) \(f(x)=\frac{2 x+1}{5} \quad\) (b) \(f(x)=3-\frac{1}{x}\) (c) \(f(x)=\sqrt{x^{3}+2} \quad\) (d) \(f(x)=(2 x-5)^{3}\) Now consider another function: $$f(x)=x^{3}+2 x+6$$ Is it possible to use the same sort of simple reversal of operations to find the inverse of this function? If so, do it. If not, explain what is different about this function that makes this task difficult.

Find a function whose graph is the given curve. The bottom half of the circle \(x^{2}+y^{2}=9\)

In Exercise 65 of Section 3.6 you were asked to solve equations in which the unknowns were functions. Now that we know about inverses and the identity function (see Exercise \(82 ),\) we can use algebra to solve such equations. For instance, to solve \(f \circ g=h\) for the unknown function \(f\) we perform the following steps: $$\begin{aligned} f \circ g &=h \\ f \circ g \circ g^{-1} &=h \circ g^{-1} \\\ f \circ I &=h \circ g^{-1} \\ f &=h \circ g^{-1} \end{aligned}$$ So the solution is \(f=h \circ g^{-1} .\) Use this technique to solve the equation \(f \circ g=h\) for the indicated unknown function. (a) Solve for \(f,\) where \(g(x)=2 x+1\) and \(h(x)=4 x^{2}+4 x+7\) (b) Solve for \(g,\) where \(f(x)=3 x+5\) and \(h(x)=3 x^{2}+3 x+2\)

Taxicab Function \(\quad\) A taxi company charges \(\$ 2.00\) for the first mile (or part of a mile) and 20 cents for each succeeding tenth of a mile (or part). Express the cost \(C\) (in dollars) of a ride as a function of the distance \(x\) traveled (in miles) for \(0 < x < 2,\) and sketch the graph of this function.

Postage Rates The domestic postage rate for first-class letters weighing 12 oz or less is 37 cents for the first ounce (or less), plus 23 cents for each additional ounce (or part of an ounce). Express the postage \(P\) as a function of the weight \(x\) of a letter, with \(0 < x \leq 12,\) and sketch the graph of this function.

When Does a Graph Represent a Function? For every integer \(n\) , the graph of the equation \(y=x^{n}\) is the graph of a function, namely \(f(x)=x^{n} .\) Explain why the graph of \(x=y^{2}\) is not the graph of a function of \(x\) . If so, of what function \(x=y^{3}\) the graph of a function of \(x ?\) If so, of what function of \(x\) is it the graph? Determine for what integers \(n\) the graph of \(x=y^{n}\) is the graph of a function of \(x .\)

Graph of the Absolute Value of a Function (a) Draw the graphs of the functions \(f(x)=x^{2}+x-6\) and \(g(x)=\left|x^{2}+x-6\right| .\) How are the graphs of \(f\) and \(g\) related? (b) Draw the graphs of the functions \(f(x)=x^{4}-6 x^{2}\) and \(g(x)=\left|x^{4}-6 x^{2}\right| .\) How are the graphs of \(f\) and \(g\) related? (c) In general, if \(g(x)=|f(x)|,\) how are the graphs of \(f\) and \(g\) related? Draw graphs to illustrate your answer.