Chapter 3

Pupil Size When the brightness x of a light source is increased, the eye reacts by decreasing the radius R of the pupil. The dependence of R on x is given by the function $$ R(x)=\sqrt{\frac{13+7 x^{0.4}}{1+4 x^{0.4}}} $$ where \(R\) is measured in millimeters and \(x\) is measured in appropriate units of brightness. (a) Find \(R(1), R(10),\) and \(R(100)\) (b) Make a table of values of \(R(x)\)

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.) $$ f(x)=4-x^{2} $$

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

\(75-82\) a Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=x^{4} $$

Relativity According to the Theory of Relativity, the length \(L\) of an object is a function of its velocity \(v\) with respect to an observer. For an object whose length at rest is \(10 \mathrm{m},\) the function is given by $$L(v)=10 \sqrt{1-\frac{v^{2}}{c^{2}}}$$ $$L(v)=10 \sqrt{1-\frac{v^{2}}{c^{2}}}$$ where \(c\) is the speed of light \((300,000 \mathrm{km} / \mathrm{s}) .\)(b) How does the length of an object change as its velocity increases?

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.) $$ g(x)=(x-1)^{2} $$

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

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=x^{3} $$

Income Tax In a certain country, income tax T is assessed according to the following function of income \(T(x)=\left\\{\begin{array}{ll}{0} & {\text { if } 0 \leq x \leq 10,000} \\\ {0.08 x} & {\text { if } 10,000

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.) $$ h(x)=(x+2)^{2} $$

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

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=x^{2}+x $$

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?

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

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=x^{4}-4 x^{2} $$

As a weather balloon is inflated, the thickness T of its rubber skin is related to the radius of the balloon by $$T(r)=\frac{0.5}{r^{2}}$$ where \(T\) and \(r\) are measured in centimeters. Graph the function \(T\) for values of \(r\) between 10 and \(100 .\)

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=x^{3}-x $$

Speeding Tickets In a certain state the maximum speed permitted on freeways is 65 mi/h, and the minimum is 40. The fine F for violating these limits is $15 for every mile above the maximum or below the minimum. (a) Complete the expressions in the following piecewise defined function, where x is the speed at which you are driving. (b) Find F(30), F(50), and F(75). (c) What do your answers in part (b) represent?

The power produced by a wind turbine depends on the speed of the wind. If a windmill has blades 3 meters long, then the power P produced by the turbine is modeled by $$P(v)=14.1 v^{3}$$ where \(P\) is measured in watts \((\mathrm{W})\) and \(v\) is measured in meters per second \((\mathrm{m} / \mathrm{s}) .\) Graph the function \(P\) for wind speeds between 1 \(\mathrm{m} / \mathrm{s}\) and 10 \(\mathrm{m} / \mathrm{s}\).

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=3 x^{3}+2 x^{2}+1 $$

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.

Fee for Service 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?

Westside Energy charges its electric customers a base rate of \(\$ 6.00\) per month, plus 10 \(\mathrm{cr}\) kilowatt-hour (kWh) for the first 300 \(\mathrm{kWh}\) used and 6 \(\mathrm{c}\) per kWh for all usage over 300 \(\mathrm{kWh}\) . Suppose a customer uses \(x \mathrm{kWh}\) of electricity in one month. (a) Express the monthly cost \(E\) as a piecewise-defined function of \(x .\) (b) Graph the function \(E\) for \(0 \leq x \leq 600\)

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ (x)=1-\sqrt[3]{x} $$

Temperature Change You place a frozen pie in an oven and bake it for an hour. Then you take the pie out and let it cool before eating it. Sketch a rough graph of the temperature of the pie as a function of time.

Toricelli's Law 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?

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 piecewise-defined function of the distance \(X\) traveled (in miles) for \(0

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=x+\frac{1}{x} $$

The domestic postage rate for first-class letters weighing 3.5 oz or less is 44 cents for the first ounce (or less), plus 17 cents for each additional ounce (or part of an ounce). Express the postage P as a piecewise-defined function of the weight \(x\) of a letter, with \(0< x \leq 3.5,\) and sketch the graph of this function.

Demand Function The amount of a commodity that is 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 \(D^{-1}(30) .\) What does your answer represent?

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 \(x .\) Is the graph of \(x=y^{3}\) 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 \(x .\)

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.

Temperature Scales 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?

In Example 7 and Exercises 82 and 83 we are given functions whose graphs consist of horizontal line segments. Such functions are often called step functions, because their graphs look like stairs. Give some other examples of step functions that arise in everyday life.

\(85-86\) . These exercises show how the graph of \(y=|f(x)|\) is obtained from the graph of \(y=f(x)\) . 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 \(f .\)

Exchange Rates The relative value of currencies fluctuates every day. When this problem was written, one Canadian dollar was worth 1.0573 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?

Sketch graphs of the functions \(f(x)=\|x\|, g(x)=[2 x \|, \text { and } h(x)=\|3 x\| \text { on separate }\) graphs. How are the graphs related? If \(n\) is a positive integer, what does the graph of \(k(x)=\|n x\|\) look like?

\(85-86\) . These exercises show how the graph of \(y=|f(x)|\) is obtained from the graph of \(y=f(x)\) . The graph of \(f(x)=x^{4}-4 x^{2}\) is shown. Use this graph to sketch the graph of \(g(x)=\left|x^{4}-4 x^{2}\right|\)

Income Tax In a certain country, the tax on incomes less than or equal to \(€ 20,000\) is 10\(\%\) . For incomes that are 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) Draw the graphs of the functions $$\begin{array}{l}{f(x)=x^{2}+x-6} \\\ {g(x)=\left|x^{2}+x-6\right|}\end{array}$$ 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.

Multiple Discounts \(A\) 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?

\(87-88\) a Sketch the graph of each function. $$ (a)f(x)=x^{3} \quad \text { (b) } g(x)=\left|x^{3}\right| $$

Pizza Cost 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?

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 \(y=\ell\) 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.

Determining When a Linear Function Has an Inverse 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?

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 \(^{\circ} \mathrm{F}\) in- stead. 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.

Finding an Inverse "in Your Head" 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 \(2^{\prime \prime}\) is "Add 2 and divide by 3 ." Use the same procedure to find the inverse of the following functions. $$ \begin{array}{ll}{\text { (a) } f(x)=\frac{2 x+1}{5}} & {\text { (b) } f(x)=3-\frac{1}{x}} \\ {\text { (c) } f(x)=\sqrt{x^{3}+2}} & {\text { (d) } f(x)=(2 x-5)^{3}}\end{array} $$ 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.

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 Identity Function The function \(I(x)=x\) is called the identity function. Show that for any function \(f\) we have \(f \circ I=f, I^{\circ} f=f,\) and \(f^{\circ} f^{-1}=f^{-1} \circ f=I .\) (This means that the identity function \(I\) behaves for functions and composition just the way the number 1 behaves for real numbers and multiplication.)

Solving an Equation for an Unknown Function In Exercise 69 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 92 ) 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 $$ h(x)=3 x^{2}+3 x+2 $$