Chapter 8

The Rent-Me Car Rental charges \(\$ 15\) per day plus \(\$ 0.22\) per mile to rent a car. Determine a linear function that can be used to calculate daily car rentals. Then use that function to determine the cost of renting a car for a day and driving 175 miles; 220 miles; 300 miles; 460 miles. See below

The period (the time required for one complete oscillation) of a simple pendulum varies directly as the square root of its length. If a pendulum 12 feet long has a period of 4 seconds, find the period of a pendulum of length 3 feet. 2 seconds

The ABC Car Rental uses the function \(f(x)=26\) for any daily use of a car up to and including 200 miles. For driving more than 200 miles per day, it uses the function \(g(x)=26+0.15(x-200)\) to determine the charges. How much would the company charge for daily driving of 150 miles? of 230 miles? of 360 miles? of 430 miles? \(\$ 26 ; \$ 30.50 ; \$ 50 ; \$ 60.50\)

Suppose the number of days it takes to complete a construction job varies inversely as the number of people assigned to the job. If it takes 7 people 8 days to do the job, how long will it take 10 people to complete the job? \(5 \frac{3}{5}\) days

Suppose that a car rental agency charges a fixed amount per day plus an amount per mile for renting a car. Heidi rented a car one day and paid \(\$ 80\) for 200 miles. On another day she rented a car from the same agency and paid \(\$ 117.50\) for 350 miles. Determine the linear function that the agency could use to determine its daily rental charges. \(f(x)=0.25 x+30\)

The number of days needed to assemble some machines varies directly as the number of machines and inversely as the number of people working. If it takes 4 people 32 days to assemble 16 machines, how many days will it take 8 people to assemble 24 machines? 24 days

If \(f(x)=3 x-2\) and \(g(x)=x^{2}+1\), find \((f \circ g)(-1)\) and \((g \circ f)(3) . \quad 4 ; 50\)

\(f(x)=\left\\{\begin{aligned} x & \text { for } x \geq 0 \\ 3 x & \text { for } x<0 \end{aligned}\right.\)

A retailer has a number of items that she wants to sell and make a profit of \(40 \%\) of the cost of each item. The function \(s(c)=c+0.4 c=1.4 c\), where \(c\) represents the cost of an item, can be used to determine the selling price. Find the selling price of items that cost \(\$ 1.50\), \(\$ 3.25, \$ 14.80, \$ 21\), and \(\$ 24.20\).

The volume of a gas at a constant temperature varies inversely as the pressure. What is the volume of a gas under a pressure of 25 pounds if the gas occupies \(15 \mathrm{cu}-\) bic centimeters under a pressure of 20 pounds?

If \(f(x)=x^{2}-2\) and \(g(x)=x+4\), find \((f \circ g)(2)\) and \((g \circ f)(-4) . \quad 34 ; 18\)

Zack wants to sell five items that cost him \(\$ 1.20, \$ 2.30\), \(\$ 6.50, \$ 12\), and \(\$ 15.60\). He wants to make a profit of \(60 \%\) of the cost. Create a function that you can use to determine the selling price of each item, and then use the function to calculate each selling price. See below

The volume \((V)\) of a gas varies directly as the temperature \((T)\) and inversely as the pressure \((P)\). If \(V=48\) when \(T=320\) and \(P=20\), find \(V\) when \(T=280\) and \(P=30 . \quad V=28\)

If \(f(x)=2 x-3\) and \(g(x)=x^{2}-3 x-4\), find \((f \circ g)(-2)\) and \((g \circ f)(1) . \quad 9 ; 0\)

"All Items \(20 \%\) Off Marked Price" is a sign at a local golf course. Create a function and then use it to determine how much one has to pay for each of the following marked items: a \(\$ 9.50\) hat, a \(\$ 15\) umbrella, a \(\$ 75\) pair of golf shoes, a \(\$ 12.50\) golf glove, a \(\$ 750\) set of golf clubs. \(f(p)=0.8 p ; \$ 7.60 ; \$ 12 ; \$ 60 ; \$ 10 ; \$ 600\)

The volume of a cylinder varies jointly as its altitude and the square of the radius of its base. If the volume of a cylinder is 1386 cubic centimeters when the radius of the base is 7 centimeters, and its altitude is 9 centimeters, find the volume of a cylinder that has a base of radius 14 centimeters if the altitude of the cylinder is 5 centimeters.

If \(f(x)=1 / x\) and \(g(x)=2 x+1\), find \((f \circ g)(1)\) and \((g \circ f)(2) . \quad \frac{1}{3} ; 2\)

\(f(x)=-2(x+1)^{3}+2\)

The linear depreciation method assumes that an item depreciates the same amount each year. Suppose a new piece of machinery costs \(\$ 32,500\) and it depreciates \(\$ 1950\) each year for \(t\) years. (a) Set up a linear function that yields the value of the machinery after \(t\) years. \(\quad f(t)=32,500-1950 t\) (b) Find the value of the machinery after 5 years. See below (c) Find the value of the machinery after 8 years. See below (d) Graph the function from part (a). (e) Use the graph from part (d) to approximate how many years it takes for the value of the machinery to become zero. (f) Use the function to determine how long it takes for the value of the machinery to become zero. $$ t=16.7 $$

The cost of labor varies jointly as the number of workers and the number of days that they work. If it costs \(\$ 900\) to have 15 people work for 5 days, how much will it cost to have 20 people work for 10 days? \(\$ 2400\)

If \(f(x)=\sqrt{x}\) and \(g(x)=3 x-1\), find \((f \circ g)(4)\) and \((g \circ f)(4) . \quad \sqrt{11} ; 5\)

\(f(x)=\left\\{\begin{aligned} 2 & \text { for } x \geq 0 \\\\-1 & \text { for } x<0 \end{aligned}\right.\)

Is \(f(x)=(3 x-2)-(2 x+1)\) a linear function? Explain your answer.

The cost of publishing pamphlets varies directly as the number of pamphlets produced. If it costs \(\$ 96\) to publish 600 pamphlets, how much does it cost to publish 800 pamphlets? \(\$ 128\)

Suppose that Bianca walks at a constant rate of 3 miles per hour. Explain what it means that the distance Bianca walks is a linear function of the time that she walks.

How would you explain the difference between direct variation and inverse variation?

Are the graphs of \(f(x)=2 \sqrt{x}\) and \(g(x)=\sqrt{2 x}\) identical? Defend your answer.

Suppose that \(y\) varies directly as the square of \(x\). Does doubling the value of \(x\) also double the value of \(y\) ? Explain your answer.

\(f(x)=x+|x|\)

Suppose that \(y\) varies inversely as \(x\). Does doubling the value of \(x\) also double the value of \(y\) ? Explain your answer.

The greatest integer function is defined by the equation \(f(x)=[x]\), where \([x]\) refers to the largest integer less than or equal to \(x\). For example, \([2.6]=2,[\sqrt{2}]=1\), \([4]=4\), and \([-1.4]=-2\). Graph \(f(x)=[x]\) for \(-4 \leq x<4\)

\(f(x)=x-|x|\)

The simple interest earned by a certain amount of money varies jointly as the rate of interest and the time (in years) that the money is invested. (a) If some money invested at \(11 \%\) for 2 years earns \(\$ 385\), how much would the same amount earn at \(12 \%\) for 1 year? \(\$ 210\) (b) If some money invested at \(12 \%\) for 3 years earns \(\$ 819\), how much would the same amount earn at \(14 \%\) for 2 years? \(\$ 637\) (c) If some money invested at \(14 \%\) for 4 years earns \(\$ 1960\), how much would the same amount earn at \(15 \%\) for 2 years? \(\$ 1050\)

Graph \(f(x)=\sqrt{x^{2}+8}, f(x)=\sqrt{x^{2}+4}\), and \(f(x)=\) \(\sqrt{x^{2}+1}\) on the same set of axes. Look at these graphs and predict the graph of \(f(x)=\sqrt{x^{2}-4}\). Now graph it with the calculator to test your prediction.

Explain the concept of a piecewise-defined function.

The period (the time required for one complete oscillation) of a simple pendulum varies directly as the square root of its length. If a pendulum 9 inches long has a period of \(2.4\) seconds, find the period of a pendulum of length 12 inches. Express the answer to the nearest tenth of a second. \(2.8\) seconds

For each of the following, predict the general shape and location of the graph, and then use your calculator to graph the function to check your prediction. (a) \(f(x)=\sqrt{x^{2}}\) (b) \(f(x)=\sqrt{x^{3}}\) (c) \(f(x)=\left|x^{2}\right|\) (d) \(f(x)=\left|x^{3}\right|\)

Is \(f(x)=\left(3 x^{2}-2\right)-(2 x+1)\) a quadratic function? Explain your answer.

\(f(x)=\frac{x}{|x|}\)

The volume of a cylinder varies jointly as its altitude and the square of the radius of its base. If the volume of a cylinder is \(549.5\) cubic meters when the radius of the base is 5 meters and its altitude is 7 meters, find the volume of a cylinder that has a base of radius 9 meters and an altitude of 14 meters. \(\quad 3560.76 \mathrm{~m}^{3}\)

Graph \(f(x)=x^{4}+x^{3}\). Now predict the graph for each of the following, and check each prediction with your graphing calculator. (a) \(f(x)=x^{4}+x^{3}-4\) (b) \(f(x)=(x-3)^{4}+(x-3)^{3}\) (c) \(f(x)=-x^{4}-x^{3}\) (d) \(f(x)=x^{4}-x^{3}\)

If \(y\) is directly proportional to \(x\) and inversely proportional to the square of \(z\), and if \(y=0.336\) when \(x=6\) and \(z=5\), find the constant of variation. \(1.4\)

Discuss whether addition, subtraction, multiplication, and division of functions are commutative operations.

Graph \(f(x)=\sqrt[3]{x}\). Now predict the graph for each of the following, and check each prediction with your graphing calculator. (a) \(f(x)=5+\sqrt[3]{x}\) (b) \(f(x)=\sqrt[3]{x+4}\) (c) \(f(x)=-\sqrt[3]{x}\) (d) \(f(x)=\sqrt[3]{x-3}-5\) (e) \(f(x)=\sqrt[3]{-x}\)

This problem is designed to reinforce ideas presented in this section. For each part, first predict the shapes and locations of the parabolas, and then use your graphing calculator to graph them on the same set of axes. (a) \(f(x)=x^{2}, f(x)=x^{2}-4, f(x)=x^{2}+1\), \(f(x)=x^{2}+5\) (b) \(f(x)=x^{2}, f(x)=(x-5)^{2}, f(x)=(x+5)^{2}\), \(f(x)=(x-3)^{2}\) (c) \(f(x)=x^{2}, f(x)=5 x^{2}, f(x)=\frac{1}{3} x^{2}, f(x)=-2 x^{2}\) (d) \(f(x)=x^{2}, f(x)=(x-7)^{2}-3, f(x)=-(x+8)^{2}+\) \(4, f(x)=-3 x^{2}-4\) (e) \(f(x)=x^{2}-4 x-2, f(x)=-x^{2}+4 x+2\), \(f(x)=-x^{2}-16 x-58, f(x)=x^{2}+16 x+58\)

Explain why the composition of two functions is not a commutative operation.

(a) Graph both \(f(x)=x^{2}-14 x+51\) and \(f(x)=x^{2}+\) \(14 x+51\) on the same set of axes. What relationship seems to exist between the two graphs? (b) Graph both \(f(x)=x^{2}+12 x+34\) and \(f(x)=x^{2}-\) \(12 x+34\) on the same set of axes. What relationship seems to exist between the two graphs? (c) Graph both \(f(x)=-x^{2}+8 x-20\) and \(f(x)=-x^{2}\) \(-8 x-20\) on the same set of axes. What relationship seems to exist between the two graphs? (d) Make a statement that generalizes your findings in parts (a) through (c).

Explain why the composition of two functions is not a commutative operation.

\(f(x)=\sqrt{3 x-4}\)

If \(f(x)=x^{2}\) and \(g(x)=\sqrt{x}\), with both having a domain of the set of nonnegative real numbers, then show that \((f \circ g)(x)=x\) and \((g \circ f)(x)=x\).