Chapter 1

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=x+3, \quad g(x)=x-3$$

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=(x-1) / 2$$

Determine whether each point lies on the graph of the equation. \(y=x^{2}-3 x+2\) (a) \(\left(\frac{5}{2}, \frac{3}{4}\right)\) (b) (-2,8)

Determine whether the relation represents \(y\) as a function of \(x .\) Explain your reasoning. $$\begin{array}{|l|l|l|l|l|l|} \hline \text { Input, } x & -3 & -1 & 0 & 1 & 3 \\ \hline \text { Output, } y & -9 & -1 & 0 & 1 & -9 \\ \hline \end{array}$$

Sketch the lines through the point with the indicated slopes on the same set of coordinate axes. Point (2,3) Slopes (a) 0 (b) 1 (c) 2 \((d)-3\)

Use a graphing utility to graph the function and estimate its domain and range. Then find the domain and range algebraically. $$f(x)=x^{2}-1$$

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{array}{l}f(x)=x^{2} \\\g(x)=\frac{1}{4} x^{2}+2 \\\h(x)=-\frac{1}{4} x^{2}\end{array}$$.

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=2 x-5, \quad g(x)=1-x$$

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=4(x-1)$$

Determine whether each point lies on the graph of the equation. \(y=\frac{1}{3} x^{3}-2 x^{2}\) (a) \(\left(2,-\frac{16}{3}\right)\) (b) (-3,9)

Determine whether the relation represents \(y\) as a function of \(x .\) Explain your reasoning. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Input, } x & 0 & 1 & 2 & 1 & 0 \\ \hline \text { Output, } y & -4 & -2 & 0 & 2 & 4 \\ \hline \end{array}$$

Sketch the lines through the point with the indicated slopes on the same set of coordinate axes. Point (-4,1) Slopes (a) 4 \((b)-2\) (c) \(\frac{1}{2}\) (d) Undefined

Use a graphing utility to graph the function and estimate its domain and range. Then find the domain and range algebraically. $$f(x)=\sqrt{x+2}$$

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{array}{l}f(x)=|x| \\\g(x)=|x|-1 \\\h(x)=3|x-3|\end{array}$$.

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=3 x^{2}, \quad g(x)=6-5 x$$

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=\sqrt[3]{x}$$

Find the slope of the line passing through the pair of points. Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points. $$(0,-10),(-4,0)$$

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=|x|\\\&g(x)=|2 x|\\\&h(x)=-2|x+2|-1\end{aligned}$$.

Use a graphing utility to graph the function and estimate its domain and range. Then find the domain and range algebraically. $$h(t)=\sqrt{4-t^{2}}$$

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=2 x+5, \quad g(x)=x^{2}-9$$

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=x^{7}$$

Find the slope of the line passing through the pair of points. Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points. $$(2,4),(4,-4)$$

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{array}{l}f(x)=-\sqrt{x} \\\g(x)=\sqrt{x+1} \\\h(x)=\sqrt{x-2}+1\end{array}$$.

Use a graphing utility to graph the function and estimate its domain and range. Then find the domain and range algebraically. $$f(x)=|x+3|$$

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=x^{2}+5, \quad g(x)=\sqrt{1-x}$$

Which sets of ordered pairs represent functions from \(A\) to \(B\) ? Explain. \(A=\\{0,1,2,3\\}\) and \(B=\\{-2,-1,0,1,2\\}\) (a) \(\\{(0,1),(1,-2),(2,0),(3,2)\\}\) (b) \(\\{(0,-1),(2,2),(1,-2),(3,0),(1,1)\\}\) (c) \(\\{(1,0),(-2,3),(-1,3),(0,0)\\}\)

Find the slope of the line passing through the pair of points. Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points. $$(-6,-1),(-6,4)$$

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=\sqrt{x}\\\&g(x)=\frac{1}{2} \sqrt{x}\\\&h(x)=-\sqrt{x-4}\end{aligned}$$.

Use a graphing utility to graph the function and estimate its domain and range. Then find the domain and range algebraically. $$f(x)=-\frac{1}{4}|x-5|$$

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=\sqrt{x^{2}-4}, \quad g(x)=\frac{x^{2}}{x^{2}+1}$$

Which sets of ordered pairs represent functions from \(A\) to \(B\) ? Explain. \(A=\\{a, b, c\\}\) and \(B=\\{0,1,2,3\\}\) (a) \(\\{(a, 1),(c, 2),(c, 3),(b, 3)\\}\) (b) \(\\{(a, 1),(b, 2),(c, 3)\\}\) (c) \(\\{(1, a),(0, a),(2, c),(3, b)\\}\)

Find the slope of the line passing through the pair of points. Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points. $$(4,9),(6,12)$$

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=\frac{1}{x}\\\&g(x)=\frac{1}{x}-2\\\&h(x)=\frac{1}{x-1}+2\end{aligned}$$.

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=\frac{1}{x}, \quad g(x)=\frac{1}{x^{2}}$$

Use the graph of the function to answer the questions. (a) Determine the domain of the function. (b) Determine the range of the function. (c) Find the value(s) of \(x\) for which \(f(x)=0\). (d) What are the values of \(x\) from part (c) referred to graphically? (e) Find \(f(0)\), if possible. (f) What is the value from part (e) referred to graphically? (g) What is the value of \(f\) at \(x=1 ?\) What are the coordinates of the point? (h) What is the value of \(f\) at \(x=-1 ?\) What are the coordinates of the point? (i) The coordinates of the point on the graph of \(f\) at which \(x=-3\) can be labeled \((-3, f(-3))\), or \((-3, )\) $$f(x)=|x-1|-2$$

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=\frac{1}{x}\\\&g(x)=\frac{1}{x}-4\\\&h(x)=\frac{1}{x+3}-1\end{aligned}$$.

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=\frac{x}{x+1}, \quad g(x)=\frac{1}{x^{3}}$$

Use the graph of the function to answer the questions. (a) Determine the domain of the function. (b) Determine the range of the function. (c) Find the value(s) of \(x\) for which \(f(x)=0\). (d) What are the values of \(x\) from part (c) referred to graphically? (e) Find \(f(0)\), if possible. (f) What is the value from part (e) referred to graphically? (g) What is the value of \(f\) at \(x=1 ?\) What are the coordinates of the point? (h) What is the value of \(f\) at \(x=-1 ?\) What are the coordinates of the point? (i) The coordinates of the point on the graph of \(f\) at which \(x=-3\) can be labeled \((-3, f(-3))\), or \((-3, )\) $$f(x)=\left\\{\begin{array}{ll} x+4, & x \leq 0 \\ 4-x^{2}, & x>0 \end{array}\right.$$

Use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) Point (3,-2) Slope \(m=0\)

Evaluate the indicated function for \(f(x)=x^{2}-1\) and \(g(x)=x-2\) algebraically. If possible, use a graphing utility to verify your answer. $$(f+g)(3)$$

Show that \(f\) and \(g\) are inverse functions algebraically. Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=x^{3}, \quad g(x)=\sqrt[3]{x}$$

Determine whether the equation represents \(y\) as a function of \(x.\) $$x^{2}+y^{2}=4$$

Use the Vertical Line Test to determine whether y is a function of x. Describe how you can use a graphing utility to produce the given graph. $$y=\frac{1}{2} x^{2}$$

Evaluate the indicated function for \(f(x)=x^{2}-1\) and \(g(x)=x-2\) algebraically. If possible, use a graphing utility to verify your answer. $$(f-g)(-2)$$

Determine whether the equation represents \(y\) as a function of \(x.\) $$x=y^{2}+1$$

Evaluate the indicated function for \(f(x)=x^{2}-1\) and \(g(x)=x-2\) algebraically. If possible, use a graphing utility to verify your answer. $$(f-g)(0)$$

Show that \(f\) and \(g\) are inverse functions algebraically. Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=\sqrt{x-4} ; \quad g(x)=x^{2}+4, \quad x \geq 0$$

Determine whether the equation represents \(y\) as a function of \(x.\) $$y=\sqrt{x^{2}-1}$$

Use the Vertical Line Test to determine whether y is a function of x. Describe how you can use a graphing utility to produce the given graph. $$0.25 x^{2}+y^{2}=1$$

Use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) Point (0,-9) Slope \(m=-2\)