Chapter 8

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=5 \sqrt{12-\sqrt[3]{x}} $$

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=0.5(x+3)^{2}+7(3+x)^{2} $$

The two functions share either an inside function or an outside function. Which is it? Describe the shared function. $$ y=(2 x+1)^{3} \text { and } y=\frac{1}{\sqrt{2 x+1}} $$

The value, \(V,\) of a car that is \(a\) years old is given by \(V=f(a)=18,000-3000 a\). Find and interpret: (a) The domain (b) The range

In Problems \(21-25,\) check that the functions are inverses. $$ f(x)=2 x-7 \text { and } g(t)=\frac{t}{2}+\frac{7}{2} $$

The cost, $$ C,\( of producing \)x\( units of a product is given by the function \)C=2000+4 x,\( up to a cost of \)\$ 10,000 .$ Find and interpret: (a) The domain (b) The range

The two functions share either an inside function or an outside function. Which is it? Describe the shared function. $$ y=\sqrt{5 x-2} \text { and } y=\sqrt{x^{2}+4} $$

Check that the functions are inverses. $$ f(x)=6 x^{7}+4 \text { and } g(t)=\left(\frac{t-4}{6}\right)^{1 / 7} $$

For what values of \(k\) does the equation \(5-3 x=k\) have a solution? What does your answer say about the range of the function \(f(x)=5-3 x ?\)

Refer to the graph of \(y=f(x)\) in Figure 8.21 . Sketch the graph of each function. \(y=f(x-3)-1\)

Check that the functions are inverses. $$ f(x)=32 x^{5}-2 \text { and } g(t)=\frac{(t+2)^{1 / 5}}{2} $$

Find a formula for \(n\) in terms of \(m\) where: \(n\) is a weight in oz and \(m\) is the weight in lbs.

For what values of \(k\) does the equation \(-1=k\) have a solution? What does your answer tell you about the range of the function \(f(x)=-1 ?\)

Evaluate and simplify \(g(0.6 c)\) given that $$ \begin{array}{l} f(x)=x^{-1 / 2} \\ g(v)=f\left(1-v^{2} / c^{2}\right) \end{array} $$

Check that the functions are inverses. $$ f(x)=\frac{x}{4}-\frac{3}{2} \text { and } g(t)=4\left(t+\frac{3}{2}\right) $$

The range of the function \(y=9-(x-2)^{2}\) is \(y \leq 9\). Find the range of the functions. $$ y=10-(x-2)^{2} $$

Find a formula for \(n\) in terms of \(m\) where: \(n\) is a length in feet and \(m\) is the length in inches.

Evaluate and simplify \(p(2)\) given that $$ \begin{aligned} V(r) &=\frac{4}{3} \pi r^{3} \\ p(t) &=V(3 t) \end{aligned} $$

Check that the functions are inverses. $$ f(x)=1+7 x^{3} \text { and } g(t)=\sqrt[3]{\frac{t-1}{7}} $$

The range of the function \(y=9-(x-2)^{2}\) is \(y \leq 9\). Find the range of the functions. $$ y=(x-2)^{2}-9 $$

Find a formula for \(n\) in terms of \(m\) where: \(n\) is a distance in \(\mathrm{km}\) and \(m\) the distance in meters.

Find a possible formula for \(f\) given that $$ \begin{aligned} f\left(x^{2}\right) &=2 x^{4}+1 \\ f(2 x) &=8 x^{2}+1 \\ f(x+1) &=2 x^{2}+4 x+3 \end{aligned} $$

Solve the equations in Problems 26-29 exactly. Use an inverse function when appropriate. $$ 2 x^{3}+7=-9 $$

The range of the function \(y=9-(x-2)^{2}\) is \(y \leq 9\). Find the range of the functions. $$ y=18-2(x-2)^{2} $$

Find a formula for \(n\) in terms of \(m\) where: \(n\) is an age in days and \(m\) the age in weeks.

Give the composition of any two functions such that (a) The outside function is a power function and the inside function is a linear function. (b) The outside function is a linear function and the inside function is a power function.

Solve the equations exactly. Use an inverse function when appropriate. $$ \frac{3 \sqrt{x}+5}{4}=5 $$

The range of the function \(y=9-(x-2)^{2}\) is \(y \leq 9\). Find the range of the functions. $$ y=\sqrt{9-(x-2)^{2}} $$

Find a formula for \(n\) in terms of \(m\) where: \(n\) is an amount in dollars and \(m\) the amount in cents.

Find (a) \(\quad f(g(x))\) (b) \(g(f(x))\) $$ f(x)=x^{3} \text { and } g(x)=5+2 x $$

Solve the equations exactly. Use an inverse function when appropriate. $$ \sqrt[3]{30-\sqrt{x}}=3 $$

A movie theater is filled to capacity with 550 people. After the movie ends, people start leaving at the rate of 100 each minute. (a) Write an expression for \(N,\) the number of people in the theater, as a function of \(t,\) the number of minutes after the movie ends. (b) For what values of \(t\) does the expression make sense in practical terms?

Find a formula for \(n\) in terms of \(m\) where: \(n\) is an elapsed time in hours and \(m\) the time in minutes.

Find (a) \(\quad f(g(x))\) (b) \(g(f(x))\) $$ f(x)=x^{3}+1 \text { and } g(x)=\sqrt{x} $$

Give the domain and range of the functions described. Let \(d=g(q)\) give the distance a certain car can travel on \(q\) gallons of gas without stopping. Its fuel economy is \(24 \mathrm{mpg},\) and its gas tank holds a maximum of 14 gallons.

Solve the equations exactly. Use an inverse function when appropriate. $$ \sqrt{x^{3}-2}=5 $$

The function \(H=f(t)\) gives the temperature, \(H^{\circ} \mathrm{F},\) of an object \(t\) minutes after it is taken out of the refrigerator and left to sit in a room. Write a new function in terms of \(f(t)\) for the temperature if: (a) The object is taken out of the refrigerator 5 minutes later. (Give a reasonable domain for your function.) (b) Both the refrigerator and the room are \(10^{\circ} \mathrm{F}\) colder.

Using \(f(t)=3 t^{2}\) and \(g(t)=2 t+1,\) find \(\begin{array}{lll}\text { (a) } & f(g(t)) \text { (b) } & g(f(t))\end{array}\) (c) \(f(f(t))\) (d) \(g(g(t))\)

Give the domain and range of the functions described. Let \(N=f(H)\) given the number of days it takes a certain kind of insect to develop as a function of the temperature \(H\left(\right.\) in \(\left.{ }^{\circ} \mathrm{C}\right)\). At \(40^{\circ} \mathrm{C}-\) the maximum it can tolerate- the insect requires 10 full days to develop. An additional day is required for every \(2^{\circ} \mathrm{C}\) drop, and it cannot develop in temperatures below \(10^{\circ} \mathrm{C}\).

In Problems \(30-34,\) find the inverse function. $$ h(x)=2 x+4 $$

The height, \(h\) in \(\mathrm{cm}\), of an eroding sand dune as a function of year, \(t,\) is given by \(h=f(t) .\) Describe the difference between this sand dune and a second one one whose height is given by (a) \(h=f(t+30)\) (b) \(h=f(t)+50\).

If \(f(g(x))=5\left(x^{2}+1\right)^{3}\) and \(g(x)=x^{2}+1\), find \(f(x)\).

Give the domain and range of the functions described. Let \(P=v(t)\) give the total amount earned (in dollars) in a week by an employee at a store as a a function of the number of hours worked. The employee earns \(\$ 7.25\) per hour and must work 4 to 6 days per week, from 7 to 9 hours per day.

Find the inverse function. $$ h(x)=9 x^{5}+7 $$

A line has equation \(y=x\) (a) Find the new equation if the line is shifted vertically up by 5 units. (b) Find the new equation if the line is shifted horizontally to the left by 5 units. (c) Compare your answers to (a) and (b) and explain your result graphically.

Give three different composite functions with the property that the outside function raises the inside function to the third power.

Give the domain and range of the functions described. Let \(T=w(r)\) give the total number of minutes of radio advertising bought in a month by a small company as a function of the rate \(r\) in \$/minute. Rates vary depending on the time of day the ad runs, ranging from $$ 40\( to $$ 100\) per minute. The company's monthly radio advertising budget is $$ 2500$.

Find the inverse function. $$ h(x)=\sqrt[3]{x+3} $$

The growth rate of a colony of bacteria at temperature \(T^{\circ} \mathrm{F}\) is \(P(T)\). The Fahrenheit temperature \(T\) for \(H^{\circ} \mathrm{C}\) is $$ T=\frac{9}{5} \cdot H+32 $$ Find an expression for \(Q(H),\) the growth rate as a function of \(H\)

Give a formula for a composite function with the property that the outside function takes the square root and the inside function multiplies by 5 and adds 2 .