Chapter 2

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=-\sqrt[3]{x}\)

Use a graphing utility to graph the function, approximate the relative minimum or maximum of the function, and estimate the open intervals on which the function is increasing or decreasing. \(f(x)=\frac{1}{4}\left(x^{4}+x^{3}-10 x^{2}+2 x-15\right)\)

Determine whether the equation represents \(y\) as a function of \(x\). \(x y-y-x-2=0\)

You are given the 2005 value of a product and the rate at which the value is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value of the product in terms of the year. (Let \(t=5\) represent 2005.) 2005 Value \(\quad\) Rate $$\$ 245,000 \quad \$$ 5600$ increase per year

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((0,-4) \quad m=-1\)

Complete the table. Use the resulting solution points to sketch the graph of the equation. \(y=5-x^{2}\) $$\begin{array}{|l|l|l|l|l|l|}\hline x & -2 & -1 & 0 & 1 & 2 \\\\\hline y & & & & & \\ \hline\end{array}$$

Find (a) \(f \circ g\), (b) \(g \circ f\), and (c) \(f \circ f\). \(f(x)=x^{2}, \quad g(x)=3 x+1\)

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=\sqrt[3]{x}-1\)

Decide whether the function is even, odd, or neither. \(f(x)=x^{6}-2 x^{2}+3\)

Fill in the blank and simplify. \(f(x)=6-4 x\) (a) \(f(3)=6-4(\quad)\) (b) \(f(-7)=6-4(\quad)\) (c) \(f(t)=6-4(\quad)\) (d) \(f(c+1)=6-(4\)

After opening the parachute, the descent of a parachutist follows a linear model. At 2:08 P.M., the height of the parachutist is 7000 feet. At \(2: 10\) P.M., the height is 4600 feet. (a) Write a linear equation that gives the height of the parachutist in terms of the time \(t\). (Let \(t=0\) represent 2:08 P.M. and let \(t\) be measured in seconds.) (b) Use the equation in part (a) to find the time when the parachutist will reach the ground.

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((-2,0)\) \(m=-4\)

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=2 x-1\)

Find (a) \(f \circ g\), (b) \(g \circ f\), and (c) \(f \circ f\). \(f(x)=x^{3}, \quad g(x)=\frac{1}{x}\)

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=\sqrt[3]{x+1}\)

Decide whether the function is even, odd, or neither. \(f(t)=t^{2}+3 t-10\)

Traveled by a Car You are driving at a constant speed. At \(4: 30\) P.M., you drive by a sign that gives the distance to Montgomery, Alabama as 84 miles. At 4:59 P.M., you drive by another sign that gives the distance to Montgomery as 56 miles. (a) Write a linear equation that gives your distance from Montgomery in terms of time \(t\). (Let \(t=0\) represent \(4: 30 \mathrm{P.M}\). and let \(t\) be measured in minutes.) (b) Use the equation in part (a) to find the time when you will reach Montgomery.

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((1,3)\) \(m=3\)

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=(x-4)(x+2)\)

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=\sqrt{x+4}, \quad g(x)=x^{2}\)

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=2-\sqrt[3]{x+1}\)

Decide whether the function is even, odd, or neither. . \(g(x)=x^{3}-5 x\)

A business purchases a piece of equipment for $$\$ 875$$. After 5 years the equipment will have no value. Write a linear equation giving the value \(V\) of the equipment during the 5 years.

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((-3,6)\) \(m=-2\)

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=x^{2}+x-2\)

Find (a) \(f \circ g\) and (b) \(g \circ f\). . \(f(x)=\sqrt[3]{x-1}, \quad g(x)=x^{3}+1\)

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=-\sqrt[3]{x-1}-4\)

Decide whether the function is even, odd, or neither. \(h(x)=x^{3}+3\)

Fill in the blank and simplify. \(f(t)=\sqrt{25-t^{2}}\) (a) \(f(3)=\sqrt{25-(})^{2}\) (b) \(f(5)=\sqrt{25-(\quad)^{2}}\) (c) \(f(x+5)=\sqrt{25-(\quad)^{2}}\) (d) \(f(2 x)=\sqrt{25-(\quad)^{2}}\)

A business purchases a piece of equipment for $$\$ 25,000$$. The equipment will be replaced in 10 years, at which time its salvage value is expected to be $$\$ 2000$$. Write a linear equation giving the value \(V\) of the equipment during the 10 years.

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((-8,3)\) \(m=-\frac{1}{2}\)

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=4-x^{2}\)

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=\frac{1}{3} x-3, \quad g(x)=3 x+1\)

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=\sqrt[3]{x+1}-1\)

Decide whether the function is even, odd, or neither. \(f(x)=x \sqrt{4-x^{2}}\)

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=2 x-3\) (a) \(f(1)\) (b) \(f(-3)\) (c) \(f(x-1)\) (d) \(f\left(\frac{1}{4}\right)\)

A store is offering a \(15 \%\) discount on all items. Write a linear equation giving the sale price \(S\) for an item with a list price \(L\).

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((4,0)\) \(m=-\frac{1}{3}\)

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=x \sqrt{x+2}\)

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=\frac{1}{2} x+1, \quad g(x)=2 x+3\)

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=2 \sqrt[3]{x-2}+1\)

Decide whether the function is even, odd, or neither. \(g(s)=4 s^{2 / 3}\)

Evaluate the function at each specified value of the independent variable and simplify. \(g(y)=7-3 y\) (a) \(g(0)\) (b) \(g\left(\frac{7}{3}\right)\) (c) \(g(s)\) (d) \(g(s+2)\)

A store is offering a \(25 \%\) discount on all shirts. Write a linear equation giving the sale price \(S\) for a shirt with a list price \(L\).

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((-2,-5)\) \(m=\frac{3}{4}\)

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=x \sqrt{x+5}\)

Find (a) \(f \circ g\) and (b) \(g \circ f\). . \(f(x)=\sqrt{x}, \quad g(x)=\sqrt{x}\)

Evaluate the function at each specified value of the independent variable and simplify. \(h(t)=t^{2}-2 t\) (a) \(h(2)\) (b) \(h(-1)\) (c) \(h(x+2)\) (d) \(h(1.5)\)

A manufacturer pays its assembly line workers $$\$ 11.50$$ per hour. In addition, workers receive a piecework rate of $$\$ 0.75$$ per unit produced. Write a linear equation for the hourly wages \(W\) in terms of the number of units \(x\) produced per hour.

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \(\begin{array}{ll}\underline{\phantom{xxx}}(6,-1) & m \text { is undefined. }\end{array}\)