Chapter 3

Find the domain of the function. $$ g(x)=\sqrt{x^{2}-2 x-8} $$

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?

A function \(f\) is given. (a) Sketch the graph of \(f .(b)\) Use the graph of \(f\) to sketch the graph of \(f^{-1} .(c)\) Find \(f^{-1} .\) $$ f(x)=3 x-6 $$

Determine whether the equation defines y as a function of x. (See Example 9.) \(x+y^{2}=9\)

Find the domain of the function. $$ f(x)=\frac{3}{\sqrt{x-4}} $$

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?

A function \(f\) is given. (a) Sketch the graph of \(f .(b)\) Use the graph of \(f\) to sketch the graph of \(f^{-1} .(c)\) Find \(f^{-1} .\) $$ f(x)=16-x^{2}, \quad x \geq 0 $$

Determine whether the equation defines y as a function of x. (See Example 9.) \(x^{2}+y=9\)

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

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 \(t(\text { in seconds). }\)

A function \(f\) is given. (a) Sketch the graph of \(f .(b)\) Use the graph of \(f\) to sketch the graph of \(f^{-1} .(c)\) Find \(f^{-1} .\) $$ f(x)=\sqrt{x+1} $$

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

Multiple Discounts You have a S50 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 \(g \circ f(x),\) 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?

Determine whether the equation defines y as a function of x. (See Example 9.) \(\sqrt{x}+y=12\)

A function \(f\) is given. (a) Sketch the graph of \(f .(b)\) Use the graph of \(f\) to sketch the graph of \(f^{-1} .(c)\) Find \(f^{-1} .\) $$ f(x)=x^{3}-1 $$

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

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 9.) \(2|x|+y=0\)

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

A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function. To evaluate \(f(x),\) divide the input by 3 and add \(\frac{2}{3}\) to the result.

Airplane Trajectory An airplane is flying at a speed of 350 \(\mathrm{milh}\) 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 \(t\)

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

A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function. To evaluate \(g(x),\) subtract 4 from the input and multiply the result by \(\frac{3}{4} .\)

Compound Interest \(\quad\) A savings account earns 5\(\%\) interest compounded annually. If you invest \(x\) dollars in such an ac- count, 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 \(A\) .

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

Determine whether the equation defines y as a function of x. (See Example 9.) \(x=y^{3}\)

A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function. Let \(T(x)\) be the amount of sales tax charged in Lemon County on a purchase of \(x\) dollars. To find the tax, take 8\(\%\) of the purchase price.

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

Determine whether the equation defines y as a function of x. (See Example 9.) \(x=y^{4}\)

A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function. Let \(V(d)\) be the volume of a sphere of diameter \(d\) . To find the volume, take the cube of the diameter, then multiply by \(\pi\) and divide by \(6 .\)

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 \(f \circ g=h\)

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 that you can make from your graphs. \(f(x)=x^{2}+c\) (a) \(c=0,2,4,6 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=0,-2,-4,-6 ; \quad[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

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

\(69-72\) . Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part \((a) ?\) Viewing rectangle \([-8,8]\) by \([-2,8]\) $$ \begin{array}{ll}{\text { (a) } y=\sqrt[4]{x}} & {\text { (b) } y=\sqrt[4]{x+5}} \\ {\text { (c) } y=2 \sqrt[4]{x+5}} & {\text { (d) } y=4+2 \sqrt[4]{x+5}}\end{array} $$

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, is \(h\) odd? What if \(g\) is odd and \(f\) is odd? What if \(g\) is odd and \(f\) is even?

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

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 that you can make from your graphs. \(f(x)=(x-c)^{2}\) (a) \(c=0,1,2,3 ; \quad[-5,5]\) 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?

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?

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 \(y=x .\) $$ f(x)=2+x $$

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 that 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?

\(69-72\) . Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part \((a) ?\) Viewing rectangle \([-4,6]\) by \([-4,4]\) $$ \begin{array}{ll}{\text { (a) } y=x^{6}} & {\text { (b) } y=\frac{1}{3} x^{6}} \\\ {\text { (c) } y=-\frac{1}{3} x^{6}} & {\text { (d) } y=-\frac{1}{3}(x-4)^{6}}\end{array} $$

Torricelli'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. Torricelli's Law gives the volume of water remaining in the tank after \(t\) minutes as $$V(t)=50\left(1-\frac{t}{20}\right)^{2} \quad 0 \leq t \leq 20$$ (a) Find \(V(0)\) and \(V(20)\) . (b) What do your answers to part (a) represent? (c) Make a table of values of \(V(t)\) for \(t=0,5,10,15,20\)

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 \(y=x .\) $$ f(x)=2-\frac{1}{2} x $$

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 that 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) \(c=1,-1,-\frac{1}{2},-2 ; \quad[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

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 \(y=x .\) $$ g(x)=\sqrt{x+3} $$

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 that 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?

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)? $$ (a)y=f(x) \quad(b) y=f(2 x) \quad(\mathbf{c}) y=f\left(\frac{1}{2} x\right) $$

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

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 \(y=x .\) $$ g(x)=x^{2}+1, \quad x \geq 0 $$

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)? $$ \begin{array}{ll}{\text { (a) } y=f(x)} & {\text { (b) } y=f(-x)} \\ {\text { (c) } y=-f(-x)} & {\text { (d) } y=f(-2 x)} \\ {\text { (e) } y=f\left(-\frac{1}{2} x\right)}\end{array} $$