Chapter 2

Graph each function in the standard viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. Over the interval specified, this function is __________. $$f(x)=-\sqrt{x} ;(0, \infty)$$

The function \(\mathrm{Y}_{2}\) is defined as \(\mathrm{Y}_{1}+k\) for some real number \(k\). Based on the two views of the graphs of \(\mathrm{Y}_{1}\) and \(\mathrm{Y}_{2}\) and the displays at the bottoms of the screens, what is the value of \(k ?\) \((-4,3)\) lies on the graph of \(Y_{1}\) First view (Graph can't copy) \((-4,8)\) lies on the graph of \(Y_{2}\) Second view (Graph can't copy)

Graph each function in the standard viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. Over the interval specified, this function is __________. $$f(x)=1-x^{3} ;(-\infty, \infty)$$

Fill in each blank with the appropriate response. (Remember that the vertical stretch or shrink factor is positive.) The graph of \(y=6 \sqrt[3]{x-3}\) can be obtained from the graph of \(y=\sqrt[3]{x}\) by shifting horizontally units to the and stretching vertically by applying a factor of

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. \(\left.y=(x-3)^{2} \text { (Exercise } 18\right)\)

Graph each function in the standard viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. Over the interval specified, this function is __________. $$f(x)=2-x^{2} ;(-\infty, 0)$$

Give the equation of each function whose graph is described. The graph of \(y=x^{2}\) is vertically shrunk by applying a factor of \(\frac{1}{2},\) and the resulting graph is shifted 7 units downward.

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. \(y=|x+4|-3\) (Exercise 21)

Graph each function in the standard viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. Over the interval specified, this function is __________. $$f(x)=|x+1| ;(-\infty,-1)$$

Give the equation of each function whose graph is described. The graph of \(y=x^{3}\) is vertically stretched by applying a factor of \(3 .\) This graph is then reflected across the \(x\) -axis. Finally, the graph is shifted 8 units upward.

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. \(y=|x-4|-3\) (Exercise 22)

Use the table to evaluate each expression, if possible. (a) \((f+g)(2)\), (b) \((f-g)(4)\), (c) \((f g)(-2)\), (d) \(\left(\frac{f}{g}\right)(0)\). $$\begin{array}{c|c|c}x & f(x) & g(x) \\\\\hline-2 & 0 & 6 \\\\\hline 0 & 5 & 0 \\\\\hline 2 & 7 & -2 \\\\\hline 4 & 10 & 5 \end{array}$$

Give the equation of each function whose graph is described. The graph of \(y=\sqrt{x}\) is shifted 3 units to the right. This graph is then vertically stretched by applying a factor of 4.5. Finally, the graph is shifted 6 units downward.

Use the table to evaluate each expression, if possible. (a) \((f+g)(2)\), (b) \((f-g)(4)\), (c) \((f g)(-2)\), (d) \(\left(\frac{f}{g}\right)(0)\). $$\begin{array}{r|c|c}\boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) & \boldsymbol{g}(\boldsymbol{x}) \\ \hline-2 & -4 & 2 \\\\\hline 0 & 8 & -1 \\\\\hline 2 & 5 & 4 \\\\\hline 4 & 0 & 0\end{array}$$

Give the equation of each function whose graph is described. The graph of \(y=\sqrt[3]{x}\) is shifted 2 units to the left. This graph is then vertically stretched by applying a factor of 1.5. Finally, the graph is shifted 8 units upward.

Checking Analytic Skills Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=\sqrt{x-3}+2$$

Without a graphing calculator, determine the domain and range of the functions. $$f(x)=(x-1)^{2}-5$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=|x+2|-3$$

Without a graphing calculator, determine the domain and range of the functions. $$f(x)=(x+8)^{2}+3$$

Why is the following not a piecewise-defined function? $$f(x)=\left\\{\begin{array}{ll} x+7 & \text { if } x \leq 4 \\ x^{2} & \text { if } x \geq 4 \end{array}\right.$$

Solve each group of equations and inequalities analytically. (a) \(|x+4|=9\) (b) \(|x+4|>9\) (c) \(|x+4|<9\)

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=\sqrt{2 x}$$

Without a graphing calculator, determine the domain and range of the functions. $$f(x)=\sqrt{x-4}$$

Solve each group of equations and inequalities analytically. (a) \(|x-3|=5\) (b) \(|x-3|>5\) (c) \(|x-3|<5\)

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=\frac{1}{2}(x+2)^{2}$$

Without a graphing calculator, determine the domain and range of the functions. $$f(x)=\sqrt{x+1}-10$$

Describe how the graph of the given function can be obtained from the graph of \(y=[x] .\) $$y=[x]-1.5$$

Solve each group of equations and inequalities analytically. (a) \(|7-2 x|=3\) (b) \(|7-2 x| \geq 3\) (c) \(|7-2 x| \leq 3\)

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=|2 x|$$

Without a graphing calculator, determine the domain and range of the functions. $$f(x)=(x-1)^{3}+4$$

Solve each group of equations and inequalities analytically. (a) \(|-9-3 x|=6\) (b) \(|-9-3 x| \geq 6\) (c) \(|-9-3 x| \leq 6\)

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=\frac{1}{2}|x|$$

Without a graphing calculator, determine the domain and range of the functions. $$f(x)=\sqrt[3]{x+7}-10$$

Describe how the graph of the given function can be obtained from the graph of \(y=[x] .\) $$y=-[x]$$

Solve each group of equations and inequalities analytically. (a) \(|2 x+1|+3=5\) (b) \(|2 x+1|+3 \leq 5\) (c) \(|2 x+1|+3 \geq 5\)

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=1-\sqrt{x}$$

Use translations of one of the basic functions \(y=x^{2}, y=x^{3},\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=(x-1)^{2}$$

Describe how the graph of the given function can be obtained from the graph of \(y=[x] .\) $$y=[x+2]$$

Solve each group of equations and inequalities analytically. (a) \(|4 x+7|+4=4\) (b) \(|4 x+7|+4>4\) (c) \(|4 x+7|+4<4\)

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=2 \sqrt{x-2}-1$$

Use translations of one of the basic functions \(y=x^{2}, y=x^{3},\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=\sqrt{x+2}$$

Use a graphing calculator in dot mode with window \([-5,5]\) by \([-3,3]\) to graph each equation. (Refer to your descriptions in Exercises 41-44.) $$y=[x]-1.5$$

Solve each group of equations and inequalities analytically. (a) \(|5-7 x|=0\) (b) \(|5-7 x| \geq 0\) (c) \(|5-7 x| \leq 0\)

Complete the table, assuming that \(f\) is en even function. $$\begin{array}{c|c|c|c|c|c|c} x & -3 & -2 & -1 & 1 & 2 & 3 \\ \hline f(x) & 21 & & -25 & & -12 & \end{array}$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=-\sqrt{1-x}$$

Use translations of one of the basic functions \(y=x^{2}, y=x^{3},\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=x^{3}+1$$

Use a graphing calculator in dot mode with window \([-5,5]\) by \([-3,3]\) to graph each equation. (Refer to your descriptions in Exercises 41-44.) $$y=[-x]$$

Solve each group of equations and inequalities analytically. (a) \(|\pi x+8|=-4\) (b) \(|\pi x+8|<-4\) (c) \(|\pi x+8|>-4\)

Complete the table, assuming that \(g\) is an odd function. $$\begin{array}{c|c|c|c|c|c|c|c} x & -5 & -3 & -2 & 0 & 2 & 3 & 5 \\ \hline g(x) & 13 & & -5 & & & -1 & \end{array}$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=\sqrt{-x}-1$$