Chapter 2

Graph each finction in the standand 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. $$f(x)=-\sqrt[3]{x},(-\infty, \infty)$$

The graph of \(y=-\frac{1}{4}|x+2|-3\) can be obtained from the graph of \(y=|x|\) by shifting horizontally _______ units to the __________ vertically shrinking by applying a factor of _______ reflecting across the __________ -axis, and shifting vertically _________ units in the direction.

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

Graph each equation by hand. $$y=3+x, y=|3+x|$$

Graph each finction in the standand 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. $$f(x)=-\sqrt{x} ;(0, \infty)$$

The graph of \(y=-\frac{2}{5}|-x|+6\) can be obtained from the graph of \(y=|x|\) by reflecting across the __________ -axis, vertically shrinking by applying a factor of __________ reflecting across the _________ -axis, and shifting vertically _________ units in the _______ direction.

Graph each equation by hand. $$y=1-x, y=|1-x|$$

Graph each finction in the standand 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. $$f(x)=1-x^{3} ;(-\infty, \infty)$$

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. $$y=x^{2}-3$$

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 byand stretching vertically by __________ applying a factor of _______.

Graph each equation by hand. $$y=3-x, y=|3-x|$$

Graph each finction in the standand 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. $$f(x)=x^{2}-2 x ;(1, \infty)$$

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. $$y=(x-3)^{2}$$

The graph of \(y=0.5 \sqrt[3]{x+2}\) can be obtained from the graph of \(y=\sqrt[3]{x}\) by shifting horizontally ____________ units to the __________ and shrinking vertically by applying a factor of _________.

Graph each equation by hand. $$y=\frac{1}{2} x, y=\left|\frac{1}{2} x\right|$$

Graph each finction in the standand 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. $$f(x)=2-x^{2} ;(-\infty, 0)$$

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

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.

Graph each equation by hand. $$y=-2 x, y=|-2 x|$$

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

Graph each finction in the standand 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. $$f(x)=|x+1| ;(-\infty,-1)$$

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

Graph each equation by hand. $$y=2 x+1, y=|2 x+1|$$

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. $$y=(x-3)^{3}$$

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) \(\left(\frac{f}{g}\right)(0)\) $$\begin{array}{r|r|r}x & f(x) & g(x) \\\\-2 & -4 & 2 \\\0 & 8 & -1 \\\2 & 5 & 4 \\\4 & 0 & 0\end{array}$$

Graph each equation by hand. $$y=3 x+3, y=|3 x+3|$$

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. $$y=(x-2)^{3}-4$$

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.

Graph each equation by hand. $$y=-2 x-4, y=|-2 x-4|$$

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

Graph each equation by hand. $$y=-3 x-2, y=|-3 x-2|$$

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

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=|x+2|-3$$

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

Graph each equation by hand. $$y=5-10 x, y=|5-10 x|$$

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

Graph each equation by hand. $$y=4-8 x, y=|4-8 x|$$

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

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=\frac{1}{2}(x+2)^{2}$$

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

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

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

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

Tables for finctions \(f\) and \(g\) are given. Evaluate cach expression, if possible. (a) \((g \circ f)(1)\)(b) \((f \circ g)(4)\)(c) \((f \circ f)(3)\) $$\begin{array}{c|c}x & f(x) \\\1 & 4 \\\2 & 3 \\\3 & 1 \\\4 & 2\end{array}\quad\quad\quad \begin{array}{c|c}x & g(x) \\\1 & 2 \\\2 & 3 \\\3 & 4 \\\4 & 5\end{array}$$

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

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. $$y=(x-1)^{2}$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=1-\sqrt{x}$$

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