Chapter 2

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(g(t)=\sqrt[3]{t-1}\)

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=\left\\{\begin{array}{ll}x^{2}+1, & x \leq 1 \\ 2 x-3, & x>1\end{array}\right.\) (a) \(f(-2)\) (b) \(f(1)\) (c) \(f\left(\frac{3}{2}\right)\) (d) \(f(0)\)

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(2 x+5=0\)

Check for symmetry with respect to both axes and the origin. \(x^{3} y=1\)

Find two functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\). (There are many correct answers.) \(h(x)=(2 x+1)^{2}\)

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=x^{3 / 2}\)

Find all real values of \(x\) such that \(f(x)=0\) \(f(x)=15-3 x\)

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(7 x+6 y-30=0\)

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=|x+2|\)

Find all real values of \(x\) such that \(f(x)=0\) \(f(x)=\frac{2 x-5}{3}\)

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(x-y-10=0\)

Check for symmetry with respect to both axes and the origin. \(y=\frac{x}{x^{2}+1}\)

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=\left\\{\begin{array}{l}x^{2}+1, x \leq 1 \\ 3 x-1, x>1\end{array}\right.\)

Find all real values of \(x\) such that \(f(x)=0\) \(f(x)=x^{2}-9\)

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(2 y-7=0\)

Check for symmetry with respect to both axes and the origin. \(x^{2}+y^{2}=25\)

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=\left\\{\begin{array}{l}2 x-1, x \leq-1 \\ x^{2}-1, x>-1\end{array}\right.\)

Find all real values of \(x\) such that \(f(x)=0\) \(f(x)=2 x^{2}-11 x+5\)

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. . \(8-5 y=0\)

Check for symmetry with respect to both axes and the origin. \(x^{2}+y^{2}=9\)

In Exercises \(49-52\), consider the graph of \(f(x)=x^{3}\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is shifted two units downward.

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=\left\\{\begin{array}{ll}x+1, & x \leq 0 \\ 4, & 02\end{array}\right.\)

Find all real values of \(x\) such that \(f(x)=0\) . \(f(x)=x^{3}-x\)

Find an equation of the line passing through the points. \((2,5),(-1,-4)\)

Use symmetry to complete the graph of the equation. \(y\) -axis symmetry \(y=-x^{2}+4\)

In Exercises \(49-52\), consider the graph of \(f(x)=x^{3}\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is shifted three units to the left.

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=\left\\{\begin{array}{ll}2 x-1, & x \leq 1 \\ 3, & 13\end{array}\right.\)

Find all real values of \(x\) such that \(f(x)=0\) \(f(x)=x^{3}-3 x^{2}-4 x+12\)

Find an equation of the line passing through the points. . \((6,-1),(-2,1)\)

Use symmetry to complete the graph of the equation. \(x\) -axis symmetry \(y=-x^{2}+4\)

In Exercises \(49-52\), consider the graph of \(f(x)=x^{3}\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is vertically stretched by a factor of 4 .

Sketch the graph of the function. \(f(x)=4-x\)

Find all real values of \(x\) such that \(f(x)=0\) \(f(x)=\frac{3}{x-1}+\frac{4}{x-2}\)

The yearly revenues (in millions of dollars) for Sonic Corporation for the years 1996 to 2005 are given by the following ordered pairs. (Source: Sonic Corporation) \((1996,151.1) \quad(1997,184.0)\) \((1998,219.1)\) \(\left(\begin{array}{lll}1999,257.6) & (2000,280.1) & (2001,330.6)\end{array}\right.\) \((2002,400.2) \quad(2003,446.6) \quad(2004,536.4)\) \((2005,623.1)\) (a) Use a graphing utility to create a scatter plot of the data. Let \(t=6\) represent 1996 . (b) Use two points on the scatter plot to find an equation of a line that approximates the data. (c) Use the regression feature of a graphing utility to find a linear model for the data. Use this model and the model from part (b) to predict the revenues in 2006 and 2007 . (d) Sonic Corporation projected its revenues in 2006 and 2007 to be \(\$ 695\) million and \(\$ 765\) million. How close are these projections to the predictions from the models? (e) Sonic Corporation also expected their yearly revenue to reach \(\$ 965\) million in 2009,2010 , or 2011 . Do the models from parts (b) and (c) support this? Explain your reasoning.

Find an equation of the line passing through the points. \((7,-4),(-7,3)\)

Use symmetry to complete the graph of the equation. Origin symmetry \(y=-x^{3}+x\)

In Exercises \(49-52\), consider the graph of \(f(x)=x^{3}\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is vertically shrunk by a factor of \(\frac{1}{3}\).

Sketch the graph of the function. \(f(x)=4 x+2\)

Find all real values of \(x\) such that \(f(x)=0\) \(f(x)=3+\frac{2}{x-1}\)

The revenues per share of stock (in dollars) for Sonic Corporation for the years 1996 to 2005 are given by the following ordered pairs. (Source: Sonic Corporation) \((1996,1.48)\) \((1997,1.90) \quad(1998,2.29)\) \(\begin{array}{lll}(1999,2.74) & (2000,3.15) & (2001,3.64)\end{array}\) \((2002,4.48)\) \((2003,5.06) \quad(2004,6.01)\) \((2005,7.00)\) (a) Use a graphing utility to create a scatter plot of the data. Let \(t=6\) represent 1996 . (b) Use two points on the scatter plot to find an equation of a line that approximates the data. (c) Use the regression feature of a graphing utility to find a linear model for the data. Use this model and the model from part (b) to predict the revenues per share in 2006 and 2007 . (d) Sonic projected the revenues per share in 2006 and 2007 to be \(\$ 8.00\) and \(\$ 8.80\). How close are these projections to the predictions from the models? (e) Sonic also expected the revenue per share to reach \(\$ 11.10\) in 2009,2010 , or 2011 . Do the models from parts (b) and (c) support this? Explain your reasoning.

Find an equation of the line passing through the points. \((4,3),(-4,-4)\)

Use symmetry to complete the graph of the equation. \(y\) -axis symmetry $$y=|x|-2$$

While driving at \(x\) miles per hour, you are required to stop quickly to avoid an accident. The distance the car travels (in feet) during your reaction time is given by \(R(x)=\frac{3}{4} x\). The distance the car travels (in feet) while you are braking is given by \(B(x)=\frac{1}{15} x^{2}\) Find the function that represents the total stopping distance \(T\). (Hint: \(T=R+B\).) Graph the functions \(R, B\), and \(T\) on the same set of coordinate axes for \(0 \leq x \leq 60\).

Consider the graph of \(f(x)=|x|\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is shifted three units to the right and two units upward.

Sketch the graph of the function. \(f(x)=x^{2}-9\)

Find the domain of the function. \(g(x)=1-2 x^{2}\)

The value (in 1982 dollars) of each dollar received by producers in each of the years from 1991 to 2005 in the United States is represented by the following ordered pairs. (Source: U.S. Bureau of Labor Statistics) \(\begin{array}{lcl}\text { ordered pairs. } & \text { Source: U.S. Bureau of Labor Statisti } \\ (1991,0.822) & (1992,0.812) & (1993,0.802) \\\ (1994,0.797) & (1995,0.782) & (1996,0.762) \\ (1997,0.759) & (1998,0.765) & (1999,0.752) \\ (2000,0.725) & (2001,0.711) & (2002,0.720) \\ (2003,0.698) & (2004,0.673) & (2005,0.642)\end{array}\) (a) Use a spreadsheet software program to generate a scatter plot of the data. Let \(t=1\) represent 1991. Do the data appear to be linear? (b) Use the regression feature of a spreadsheet software program to find a linear model for the data. (c) Use the model to estimate the value (in 1982 dollars) of 1 dollar received by producers in 2007 and in 2008 . Discuss the reliability of your estimates based on your scatter plot and the graph of your linear model for the data.

Find an equation of the line passing through the points. \((-9,11),(-9,14)\)

The weekly cost \(C\) of producing \(x\) units in a manufacturing process is given by the function \(C(x)=70 x+800\) The number of units \(x\) produced in \(t\) hours is given by \(x(t)=40 t\) Find and interpret \((C \circ x)(t)\).

Consider the graph of \(f(x)=|x|\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is reflected in the \(x\) -axis, shifted two units to the left, and shifted three units upward.