Chapter 2

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=-x \text { and } g(x)=-x$$

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$x+y=16$$

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,-3) \text { and }(4,1) $$

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)=\frac{1}{2}(x-1)^{2} $$

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(3,-4)\( and \)(3,5)$$

Find \(f+g, f-g, f g,\) and \(\frac{f}{g}\). Determine the domain for each function. $$f(x)=6 x^{2}-x-1, g(x)=x-1$$

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=\sqrt[3]{x-4} \text { and } g(x)=x^{3}+4$$

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$x+y=25$$

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,-2) \text { and }(4,3) $$

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)=\sqrt{x}+2 $$

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=2,\) passing through \((3,5)\)

Find \(f+g, f-g, f g,\) and \(\frac{f}{g}\). Determine the domain for each function. $$f(x)=\sqrt{x}, g(x)=x-4$$

The functions in Exercises \(11-30\) are all one-to-one. For each function: a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=x+3$$

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$x^{2}+y=16$$

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (3.5,8.2) \text { and }(-0.5,6.2) $$

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)=\sqrt{x}+1 $$

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=4,\) passing through \((1,3)\)

Find \(f+g, f-g, f g,\) and \(\frac{f}{g}\). Determine the domain for each function. $$f(x)=\frac{1}{x}, g(x)=x-5$$

The functions in Exercises \(11-30\) are all one-to-one. For each function: a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=x+5$$

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$x^{2}+y=25$$

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,-\sqrt{3}) \text { and }(\sqrt{5}, 0) $$

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)=\sqrt{x+2} $$

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=6,\) passing through \((-2,5)\)

Find \(f+g, f-g, f g,\) and \(\frac{f}{g}\). Determine the domain for each function. $$f(x)=2+\frac{1}{x}, g(x)=\frac{1}{x}$$

The functions in Exercises \(11-30\) are all one-to-one. For each function: a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=2 x$$

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$x^{2}+y^{2}=16$$

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,-\sqrt{3}) \text { and }(\sqrt{5}, 0) $$

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)=\sqrt{x+1} $$

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=8,\) passing through \((4,-1)\)

Find \(f+g, f-g, f g,\) and \(\frac{f}{g}\). Determine the domain for each function. $$f(x)=6-\frac{1}{x}, g(x)=\frac{1}{x}$$

The functions in Exercises \(11-30\) are all one-to-one. For each function: a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=4 x$$

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$x^{2}+y^{2}=25$$

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,-\sqrt{2}) \text { and }(\sqrt{7}, 0) $$

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)=-\sqrt{x+2} $$

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-3,\) passing through \((-2,-3)\)

Find \(f+g, f-g, f g,\) and \(\frac{f}{g}\). Determine the domain for each function. $$f(x)=\sqrt{x+4}, g(x)=\sqrt{x-1}$$

The functions in Exercises \(11-30\) are all one-to-one. For each function: a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=2 x+3$$

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$x=y^{2}$$

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (3 \sqrt{3}, \sqrt{5}) \text { and }(-\sqrt{3}, 4 \sqrt{5}) $$

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)=-\sqrt{x+1} $$

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-5,\) passing through \((-4,-2)\)

Find \(f+g, f-g, f g,\) and \(\frac{f}{g}\). Determine the domain for each function. $$f(x)=\sqrt{x+6}, g(x)=\sqrt{x-3}$$

The functions in Exercises \(11-30\) are all one-to-one. For each function: a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=3 x-1$$

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (2 \sqrt{3}, \sqrt{6}) \text { and }(-\sqrt{3}, 5 \sqrt{6}) $$

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)=\sqrt{-x+2} $$

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-4,\) passing through \((-4,0)\)

Find: a. \((f \circ g)(x)\) b. \(\left(g^{\circ} f\right)(x)\) c. \((f \circ g)(2)\) $$f(x)=2 x, g(x)=x+7$$

The functions in Exercises \(11-30\) are all one-to-one. For each function: a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=x^{3}+2$$

In Exercises \(9-20,\) determine whether each equation defines \(y\) as a function of \(x .\) $$y=\sqrt{x+4}$$

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ \left(\frac{7}{3}, \frac{1}{5}\right) \text { and }\left(\frac{1}{3}, \frac{6}{5}\right) $$