Chapter 3

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

Area of a Ripple A stone is dropped in a lake, creating a circular ripple that travels outward at a speed of 60 \(\mathrm{cm} / \mathrm{s}\) . (a) Find a function \(g\) that models the radius as a function of time. (b) Find a function \(f\) that models the area of the circle as a function of the radius. (c) Find \(f \circ g .\) What does this function represent?

\(51-58=\) Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$ V(x)=\frac{1-x^{2}}{x^{3}} $$

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. \(f(x)=\frac{x+12}{x-6}\)

Find the domain of the function. $$ f(x)=\frac{(x+1)^{2}}{\sqrt{2 x-1}} $$

Inflating a Balloon A spherical balloon is being inflated. The radius of the balloon is increasing at the rate of 1 \(\mathrm{cm} / \mathrm{s}\) . (a) Find a function \(f\) that models the radius as a function of time. (b) Find a function \(g\) that models the volume as a function of the radius. (c) Find \(g \circ f .\) What does this function represent?

\(51-58=\) Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$ V(x)=\frac{1}{x^{2}+x+1} $$

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. \(f(x)=\sqrt{x^{3}-4 x+1}\)

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

Area of a Balloon A spherical weather balloon is being inflated. The radius of the balloon is increasing at the rate of 2 \(\mathrm{cm} / \mathrm{s}\) . Express the surface area of the balloon as a function of time \(t\) (in seconds).

Height of a Ball If a ball is thrown directly upward with a velocity of 40 \(\mathrm{ft} / \mathrm{s}\) , its height (in feet) after \(t\) seconds is given by \(y=40 t-16 t^{2} .\) What is the maximum height attained by the ball?

If \(f(x)=\sqrt{2 x-x^{2}}\) graph the following functions in the viewing rectangle \([-5,5]\) by \([-4,4]\) , How is each graph related to the graph in part (a)? \(\begin{array}{llll}{\text { (a) } y=f(x)} & {\text { (b) } y=f(2 x)} & {\text { (c) } y=f\left(\frac{1}{2} x\right)}\end{array}\)

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. \(f(x)=|x|-|x-6|\)

Production Cost The cost \(C\) in dollars of producing \(x\) yards of a certain fabric is given by the function $$ C(x)=1500+3 x+0.02 x^{2}+0.0001 x^{3} $$ (a) Find \(C(10)\) and \(C(100) .\) (b) What do your answers in part (a) represent? (c) Find \(C(0)\) . (This number represents the fixed costs.)

Multiple Discounts You have a \(\$ 50\) coupon from the manufacturer good for the purchase of a cell phone. The store where you are purchasing your cell phone is offering a 20\(\%\) discount on all cell phones. Let \(x\) represent the regular price of the cell phone. (a) Suppose only the 20\(\%\) discount applies. Find a function \(f\) that models the purchase price of the cell phone as a function of the regular price \(x .\) (b) Suppose only the \(\$ 50\) coupon applies. Find a function \(g\) that models the purchase price of the cell phone as a function of the sticker price \(x .\) (c) If you can use the coupon and the discount, then the purchase price is either \(f \circ g(x)\) or \(g\) o \(f(x),\) depending on the order in which they are applied to the price. Find both \(f \circ g(x)\) and \(g \circ f(x) .\) Which composition gives the lower price?

If \(f(x)=\sqrt{2 x-x^{2}}\) graph the following functions in the viewing rectangle \([-5,5]\) by \([-4,4]\) , How is each graph related to the graph in part (a)? \(\begin{array}{llll}{\text { (a) } y=f(x)} & {\text { (b) } y=f(- x)} & {\text { (c) } y=-f\left(-x\right)}\\\ {\text { (d) } y=f(- 2x)} & {\text { (e) } y=f\left(-\frac{1}{2} x\right)}\end{array}\)

Path of a Ball \(A\) ball is thrown across a playing field. Its path is given by the equation \(y=-0.005 x^{2}+x+5\) where \(x\) is the distance the ball has traveled horizontally, and \(y\) is its height above ground level, both measured in feet. (a) What is the maximum height attained by the ball? (b) How far has it traveled horizontally when it hits the ground?

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. \(f(x)=x \cdot|x|\)

Area of a Sphere The surface area \(S\) of a sphere is a function of its radius \(r\) given by $$S(r)=4 \pi r^{2}$$ (a) Find \(S(2)\) and \(S(3)\) . (b) What do your answers in part (a) represent?

Airplane Trajectory An airplane is flying at a speed of 350 \(\mathrm{mi} / \mathrm{h}\) at an altitude of one mile. The plane passes directly above a radar station at time \(t=0\) . (a) Express the distance \(s\) (in miles) between the plane and the radar station as a function of the horizontal distance \(d\) (in miles) that the plane has flown. (b) Express \(d\) as a function of the time \(t\) (in hours) that the plane has flown. (c) Use composition to express \(s\) as a function of \(t .\)

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = x^2$$

Determine whether the equation defines y as a function of x. (See Example 10.) $$ x^{2}+2 y=4 $$

Revenue A manufacturer finds that the revenue generated by selling \(x\) units of a certain commodity is given by the function \(R(x)=80 x-0.4 x^{2},\) where the revenue \(R(x)\) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x\). \(f(x)=2+x\)

Multiple Discounts An appliance dealer advertises a 10\(\%\) discount on all his washing machines. In addition, the manufacturer offers a \(\$ 100\) rebate on the purchase of a washing machine. Let \(x\) represent the sticker price of the washing machine. (a) Suppose only the 10\(\%\) discount applies. Find a function \(f\) that models the purchase price of the washer as a function of the sticker price \(x .\) (b) Suppose only the \(\$ 100\) rebate applies. Find a function \(g\) that models the purchase price of the washer as a function of the sticker price \(x .\) (c) Find \(f \circ g\) and \(g \circ f .\) What do these functions represent? Which is the better deal?

Determine whether the equation defines y as a function of x. (See Example 10.) $$ 3 x+7 y=21 $$

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = x^{-3}$$

Sales A soft-drink vendor at a popular beach analyzes his sales records, and finds that if he sells \(x\) cans of soda pop in one day, his profit (in dollars) is given by $$ P(x)=-0.001 x^{2}+3 x-1800 $$ What is his maximum profit per day, and how many cans must he sell for maximum profit?

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x\). \(f(x)=2-\frac{1}{2} x\)

TTorricelli's Law A tank holds 50 gallons of water, which drains from a leak at the bottom, causing the tank to empty in 20 minutes. The tank drains faster when it is nearly full because the pressure on the leak is greater. Torricelly's Law gives the volume of water remaining in the tank after t minutes as $$ V(t)=50\left(1-\frac{t}{20}\right)^{2} \quad 0 \leq t \leq 20 $$ ($$ \begin{array}{l}{\text { (a) Find } V(0) \text { and } V(20) .} \\ {\text { (b) What do your answers to part (a) represent? }} \\ {\text { (c) Make a table of values of } V(t) \text { for } t=0,5,10,15,20 \text { . }}\end{array} $$

Compound Interest \(\quad\) A savings account earns 5\(\%\) interest compounded annually. If you invest \(x\) dollars in such an account, then the amount \(A(x)\) of the investment after one year is the initial investment plus \(5 \% ;\) that is, \(A(x)=x+0.05 x=1.05 x .\) Find $$\begin{array}{l}{A \circ A} \\ {A \circ A \circ A} \\ {A \circ A \circ A \circ A}\end{array}$$ What do these compositions represent? Find a formula for what you get when you compose \(n\) copies of \(A .\)

Determine whether the equation defines y as a function of x. (See Example 10.) $$ x=y^{2} $$

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = x^{2} + {x}$$

Advertising The effectiveness of a television com- mercial depends on how many times a viewer watches it. After some experiments an advertising agency found that if the effectiveness \(E\) is measured on a scale of 0 to \(10,\) then $$ E(n)=\frac{2}{3} n-\frac{1}{90} n^{2} $$ where \(n\) is the number of times a viewer watches a given commercial. For a commercial to have maximum effectiveness, how many times should a viewer watch it?

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x\). \(g(x)=\sqrt{x+3}\)

Blood Flow As blood moves through a vein or an artery, its velocity \(v\) is greatest along the central axis and decreases as the distance \(r\) from the central axis increases (see the figure). The formula that gives \(v\) as a function of \(r\) is called the law of laminar flow. For an artery with radius \(0.5 \mathrm{cm},\) we have $$ v(r)=18,500\left(0.25-r^{2}\right) \quad 0 \leq r \leq 0.5 $$ $$ \begin{array}{l}{\text { (a) Find } v(0.1) \text { and } v(0.4) .} \\ {\text { (b) What do your answers to part (a) tell you about the flow }} \\ {\text { of blood in this artery? }} \\ {\text { (c) Make a table of values of } v(r) \text { for } r=0,0.1,0.2,0.3,} \\ {0.4,0.5 .}\end{array} $$

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = x^{4} - {4x}^{2}$$

Pharmaceuticals \(\quad\) When a certain drug is taken orally, the concentration of the drug in the patient's bloodstream after \(t\) minutes is given by \(C(t)=0.06 t-0.0002 t^{2},\) where \(0 \leq t \leq 240\) and the concentration is measured in \(\mathrm{mg} / \mathrm{L}\) . When is the maximum serum concentration reached, and what is that maximum concentration?

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x\). \(g(x)=x^{2}+1, \quad x \geq 0\)

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 $$ R(x)=\sqrt{\frac{13+7 x^{0.4}}{1+4 x^{0.4}}} $$ (a) Find \(R(1), R(10),\) and \(R(100)\) (b) Make a table of values of \(R(x)\)

Solving an Equation for an Unknown Function suppose that $$\begin{aligned} g(x) &=2 x+1 \\ h(x) &=4 x^{2}+4 x+7 \end{aligned}$$ Find a function \(f\) such that \(f \circ g=h .\) (Think about what operations you would have to perform on the formula for \(g\) to end up with the formula for \(h . )\) Now suppose that $$\begin{array}{l}{f(x)=3 x+5} \\ {h(x)=3 x^{2}+3 x+2}\end{array}$$ Use the same sort of reasoning to find a function \(g\) such that \(f \circ g=h .\)

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = x^{3} - {x}$$

Agriculture The number of apples produced by each tree in an apple orchard depends on how densely the trees are planted. If \(n\) trees are planted on an acre of land, then each tree produces \(900-9 n\) apples. So the number of apples produced per acre is $$ A(n)=n(900-9 n) $$ How many trees should be planted per acre in order to obtain the maximum yield of apples?

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 $$ L(v)=10 \sqrt{1-\frac{v^{2}}{c^{2}}} $$ where \(c\) is the speed of light. $$ \begin{array}{l}{\text { (a) Find } L(0.5 c), L(0.75 c), \text { and } L(0.9 c)} \\ {\text { (b) How does the length of an object change as its velocity }} \\ {\text { increases? }}\end{array} $$

Compositions of Odd and Even Functions Suppose that $$h=f \circ g$$ If \(g\) is an even function, is \(h\) necessarily even? If \(g\) is odd and \(f\) \(h\) odd? What if \(g\) is odd and \(f\) is odd? What if \(g\) is odd and \(f\) is even?

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = 3x^{3} + {2x}^{2} + {1}$$

Migrating Fish A fish swims at a speed \(v\) relative to the water, against a current of 5 mi/h. Using a mathematical model of energy expenditure, it can be shown that the total energy \(E\) required to swim a distance of 10 \(\mathrm{mi}\) is given by $$ E(v)=2.73 v^{3} \frac{10}{v-5} $$ Biologists believe that migrating fish try to minimize the total energy required to swim a fixed distance. Find the value of \(v\) that minimizes energy required. NOTE This result has been verified; migrating fish swim against a current at a speed 50\(\%\) greater than the speed of the current.

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?

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = {1} -{3}\sqrt{x}$$

Highway Engineering A highway engineer wants to estimate the maximum number of cars that can safely travel a particular highway at a given speed. She assumes that each car is 17 \(\mathrm{ft}\) long, travels at a speed \(s,\) and follows the car in front of it at the "safe following distance" for that speed. She finds that the number \(N\) of cars that can pass a given point per minute is modeled by the function $$ N(s)=\frac{88 s}{17+17\left(\frac{s}{20}\right)^{2}} $$ At what speed can the greatest number of cars travel the highway safely?