Chapter 2

Skills Graph each piecewise-defined function in Exercises \(9-20 .\) Is \(f\) continuous on its domain? Do not use a calculator. $$f(x)=\left\\{\begin{array}{ll} -2 x & \text { if } x<-3 \\ 3 x-1 & \text { if }-3 \leq x \leq 2 \\ -4 x & \text { if } x>2 \end{array}\right.$$

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=|x|, \quad y_{2}=2|x|, \quad y_{3}=2.5|x|$$

Give a short answer to each question. If the range of \(y=f(x)\) is \([-2, \infty),\) what is the range of \(y=|f(x)| ?\)

Write the equation that results in the desired translation. Do not use a calculator. The squaring function, shifted 1000 units to the left and 255 units downward

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$\left(\frac{f}{g}\right)(-1)$$

Skills Graph each piecewise-defined function in Exercises \(9-20 .\) Is \(f\) continuous on its domain? Do not use a calculator. $$f(x)=\left\\{\begin{array}{ll} -\frac{1}{2} x^{2}+2 & \text { if } x \leq 2 \\\ \frac{1}{2} x & \text { if } x>2 \end{array}\right.$$

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=\sqrt[3]{x}, \quad y_{2}=-\sqrt[3]{x}, \quad y_{3}=-2 \sqrt[3]{x}$$

Give a short answer to each question. If the range of \(y=f(x)\) is \((-\infty,-2],\) what is the range of \(y=|f(x)| ?\)

Write the equation that results in the desired translation. Do not use a calculator. Explain how the graph of \(g(x)=f(x)+4\) is obtained from the graph of \(y=f(x)\)

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$\left(\frac{f}{g}\right)(4)$$

Skills Graph each piecewise-defined function in Exercises \(9-20 .\) Is \(f\) continuous on its domain? Do not use a calculator. $$f(x)=\left\\{\begin{array}{ll} x^{3}+5 & \text { if } x \leq 0 \\ -x^{2} & \text { if } x>0 \end{array}\right.$$

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=x^{2}, \quad y_{2}=(x-2)^{2}+1, \quad y_{3}=-(x+2)^{2}$$

Give a short answer to each question. Why can't the range of \(y=|f(x)|\) include \(-1,\) for any function \(f ?\)

Write the equation that results in the desired translation. Do not use a calculator. Explain how the graph of \(g(x)=f(x+4)\) is obtained from the graph of \(y=f(x)\)

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(f-g)(2)$$

Skills Graph each piecewise-defined function in Exercises \(9-20 .\) Is \(f\) continuous on its domain? Do not use a calculator. $$f(x)=\left\\{\begin{array}{ll} 2 x & \text { if }-5 \leq x<-1 \\ -2 & \text { if }-1 \leq x<0 \\ x^{2}-2 & \text { if } 0 \leq x \leq 2 \end{array}\right.$$

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=|x|, \quad y_{2}=-2|x-1|+1, \quad y_{3}=-\frac{1}{2}|x|-4$$

Use graphing to determine the domain and range of \(y=f(x)\) and of \(y=|f(x)|\). $$f(x)=(x+1)^{2}-2$$

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(f-g)(-2)$$

Skills Graph each piecewise-defined function in Exercises \(9-20 .\) Is \(f\) continuous on its domain? Do not use a calculator. $$f(x)=\left\\{\begin{array}{ll} 0.5 x^{2} & \text { if }-4 \leq x \leq-2 \\\ x & \text { if }-2

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=\sqrt{x}, \quad y_{2}=-\sqrt{x}, \quad y_{3}=\sqrt{-x}$$

Use graphing to determine the domain and range of \(y=f(x)\) and of \(y=|f(x)|\). $$f(x)=2-\frac{1}{2} x$$

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(g-f)(-2)$$

Skills Graph each piecewise-defined function in Exercises \(9-20 .\) Is \(f\) continuous on its domain? Do not use a calculator. $$f(x)=\left\\{\begin{array}{ll} x^{3}+3 & \text { if }-2 \leq x \leq 0 \\\ x+3 & \text { if } \quad 0

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=x^{2}-1, \quad y_{2}=\left(\frac{1}{2} x\right)^{2}-1, \quad y_{3}=(2 x)^{2}-1$$

Use graphing to determine the domain and range of \(y=f(x)\) and of \(y=|f(x)|\). $$f(x)=-1-(x-2)^{2}$$

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$\left(\frac{f}{g}\right)\left(\frac{1}{2}\right)$$

Skills Graph each piecewise-defined function in Exercises \(9-20 .\) Is \(f\) continuous on its domain? Do not use a calculator. $$f(x)=\left\\{\begin{array}{ll} -2 x & \text { if }-3 \leq x<-1 \\ x^{2}+1 & \text { if }-1 \leq x \leq 2 \\ \frac{1}{2} x^{3}+1 & \text { if } 2

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=3-|x|, \quad y_{2}=3-|3 x|, \quad y_{3}=3-\left|\frac{1}{3} x\right|$$

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$\left(\frac{g}{f}\right)(0)$$

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=\sqrt[3]{x}, \quad y_{2}=\sqrt[3]{-x}, \quad y_{3}=\sqrt[3]{-(x-1)}$$

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=\sqrt[3]{x}, \quad y_{2}=2-\sqrt[3]{x}, \quad y_{3}=1+\sqrt[3]{x}$$

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=4 x-1, g(x)=6 x+3$$ (c) \(\left(\begin{array}{l}f \\ g\end{array}\right)(x)=\frac{4 x-1}{6 x+3} ;\) domain: \(\left(-\infty,-\frac{1}{2}\right) \cup\left(-\frac{1}{2}, \infty\right)\) (d) \((f \circ g)(x)=24 x+11 ;\) domain: \((-\infty, \infty)\) (e) \((g \circ f)(x)=24 x-3 ;\) domain: \((-\infty, \infty)\)

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 __________. $$f(x)=x^{5} ;(-\infty, \infty)$$

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=9-2 x, g(x)=-5 x+2$$

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 __________. $$f(x)=-x^{3} ;(-\infty, \infty)$$

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=|x+3|, g(x)=2 x$$

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 __________. $$f(x)=x^{4} ;(-\infty, 0)$$

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=|2 x-4|, g(x)=x+1$$

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 __________. $$f(x)=x^{4} ;(0, \infty)$$

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=\sqrt[3]{x+4}, g(x)=x^{3}+5$$

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 __________. $$f(x)=-|x| ;(-\infty, 0)$$

Fill in each blank with the appropriate response. (Remember that the vertical stretch or shrink factor is positive.) The graph of \(y=-4 x^{2}\) can be obtained from the graph of \(y=x^{2}\) by vertically stretching by applying a factor of_______________ and reflecting across the _________-axis.

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=\sqrt[3]{6-3 x}, g(x)=2 x^{3}+1$$

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 __________. $$f(x)=-|x| ;(0, \infty)$$

Fill in each blank with the appropriate response. (Remember that the vertical stretch or shrink factor is positive.) The graph of \(y=-6 \sqrt{x}\) can be obtained from the graph of \(y=\sqrt{x}\) by vertically stretching by applying a factor of__________and reflecting across the_______-axis.

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=\sqrt{x^{2}+3}, g(x)=x+1$$

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 __________. $$f(x)=-\sqrt[3]{x} ;(-\infty, \infty)$$

Concept Check 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 \(k ?\) \((6,2)\) lies on the graph of \(Y_{1}\) First view (Graph can't copy) \((6,-1)\) lies on the graph of \(Y_{2}\) Second view (Graph can't copy)

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=\sqrt{2+4 x^{2}}, g(x)=x$$