Chapter 2

Use the point on the line and the slope of the line to find three additional points through which the line passes. \(\begin{array}{ll}\text { Point } & \text { Slope }\end{array}\) \((5,-2) \quad m=0\)

Find \(x\) such that the distance between the points is 15 . \((3,-4),(x, 5)\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f g)(-6)\)

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(f(x)=x^{2 / 3}\)

Determine whether the equation represents \(y\) as a function of \(x\). . \(x=y^{2}\)

An item that sells for $$\$ 145.99$$ has a sales tax of $$\$ 10.22$$ (a) Find a mathematical model that gives the amount of sales tax \(y\) in terms of the retail price \(x\). (b) Use the model to find the sales tax on a purchase that has a retail price of \(\$ 540.50\).

Use the point on the line and the slope of the line to find three additional points through which the line passes. \(\begin{array}{ll}\text { Point } & \text { Slope }\end{array}\) \((-3,4)\) \(m=0\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \(\left(\frac{f}{g}\right)(5)\)

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

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(y=x \sqrt{x+3}\)

Determine whether the equation represents \(y\) as a function of \(x\). \(x^{2}+y=4\)

Find \(y\) such that the distance between the points is 20 . \((-15, y),(-3,-7)\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \(\left(\frac{f}{g}\right)(0)\)

Describe the sequence of transformations from \(f(x)=\sqrt{x}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. . \(g(x)=\sqrt{x+5}-2\)

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(y=|x+1|+|x-1|\)

Determine whether the equation represents \(y\) as a function of \(x\). \(x+y^{2}=4\)

You are buying gasoline and notice that 14 gallons of gasoline is the same as 53 liters. (a) Use this information to find a mathematical model that relates gallons to liters. (b) Use the model to complete the table. $$\begin{array}{|l|l|l|l|l|l|}\hline \text { Gallons } & 5 & 10 & 20 & 25 & 30 \\\ \hline \text { Liters } & & & & & \\\\\hline\end{array}$$

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f-g)(0)\)

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. \(f(x)=x^{2}-4 x+1\)

Determine whether the equation represents \(y\) as a function of \(x\). \(2 x+3 y=4\)

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 . $$\$ 2540 \quad \$ 125$$ increase per year

Use the point on the line and the slope of the line to find three additional points through which the line passes. \(\begin{array}{ll}\text { Point } & \text { Slope }\end{array}\) \((5,-6)\) \(m=1\)

Determine whether each point is a solution of the equation. Equation Points . \(2 x-3 y+11=0\) (a) \((2,5)\) (b) \((3,2)\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f+g)(1)\)

Describe the sequence of transformations from \(f(x)=\sqrt{x}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=\sqrt{2 x}-5\)

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. \(f(x)=-x^{2}+6 x+3\)

Determine whether the equation represents \(y\) as a function of \(x\). \(x^{2}+y^{2}-2 x-4 y+1=0\)

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 $$156 $$quad $$ 4.50$$ increase per year

Use the point on the line and the slope of the line to find three additional points through which the line passes. \(\begin{array}{ll}\text { Point } & \text { Slope }\end{array}\) \((10,-6)\) \(m=-1\)

Determine whether each point is a solution of the equation. Equation Points \(y=2 x^{2}-7 x+3\) (a) \((1,-1)\) (b) \((3,0)\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \(\left(\frac{f}{g}\right)(-1)-g(3)\)

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

Determine whether the equation represents \(y\) as a function of \(x\). \(y^{2}=x^{2}-1\)

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 $$\$ 20,400 \quad \$ 2000$$ decrease per year

Use the point on the line and the slope of the line to find three additional points through which the line passes. \(\begin{array}{ll}\text { Point } & \text { Slope }\end{array}\) \((-6,-1)\) \(m=\frac{1}{2}\)

Determine whether each point is a solution of the equation. Equation Points \(y=\sqrt{x-5}\) (a) \((9,2)\) (b) \((21,4)\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((2 f)(5)+(3 g)(-4)\)

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

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

Determine whether the equation represents \(y\) as a function of \(x\). \(y=\sqrt{x+5}\)

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 $$\$ 45,000 \quad \$ 2800$$ decrease per year

Use the point on the line and the slope of the line to find three additional points through which the line passes. \(\begin{array}{ll}\text { Point } & \text { Slope }\end{array}\) \((7,-5)\) \(m=-\frac{2}{3}\)

Determine whether each point is a solution of the equation. Equation Points \(y=\frac{x+1}{5-x}\) (a) \(\left(1, \frac{1}{2}\right)\) (b) \((0,1)\)

Find (a) \(f \circ g\), (b) \(g \circ f\), and (c) \(f \circ f\). \(f(x)=3 x, \quad g(x)=2 x+5\)

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. \(f(x)=\frac{1}{4}\left(-4 x^{4}-5 x^{3}+10 x^{2}+8 x+6\right)\)

Determine whether the equation represents \(y\) as a function of \(x\). \(x^{2} y-x^{2}+4 y=0\)

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 $$\$ 154,000 \quad \$ 12,500$$ 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 \(\begin{array}{ll}\text { 25. }(7,0) & m=1\end{array}\) \((0,-4) \quad m=-1\)

Complete the table. Use the resulting solution points to sketch the graph of the equation. \(y=\frac{3}{4} x-1\) $$\begin{array}{|l|l|l|l|l|l|}\hline x & -2 & 0 & 1 & \frac{4}{3} & 2 \\\\\hline y & & & & & \\ \hline\end{array}$$

Find (a) \(f \circ g\), (b) \(g \circ f\), and (c) \(f \circ f\). \(f(x)=2 x-1, \quad g(x)=7-x\)