Chapter 3

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=-x^{2}+2 x+5\)

Use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window. Zoom out far enough to see the right-hand and left-hand behavior of each graph. Do the graphs of \(f\) and \(g\) have the same right-hand and Ieft- hand behavior? Explain why or why not. \(f(x)=-2 x^{3}+4 x^{2}-1, \quad g(x)=2 x^{3}\)

(a) find the domain of the function, (b) decide whether the function is continuous, and (c) identify any horizontal and vertical asymptotes. Verify your answer to part (a) both graphically by using a graphing utility and numerically by creating a table of values. $$f(x)=\frac{4 x^{3}-x^{2}+3}{3 x^{3}+24}$$

Find (a) \(f \circ g\) and (b) \(g \circ f\). $$f(x)=5 x+8, g(x)=2 x^{2}-1$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{2 x}{x^{2}+x-2}$$

Use synthetic division to divide. $$\left(5 x^{3}+6 x+8\right) \div(x+2)$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=x^{4}+29 x^{2}+100$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=-x^{2}-4 x+1\)

Use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window. Zoom out far enough to see the right-hand and left-hand behavior of each graph. Do the graphs of \(f\) and \(g\) have the same right-hand and Ieft- hand behavior? Explain why or why not. \(f(x)=-\left(x^{4}-6 x^{2}-x+10\right), \quad g(x)=x^{4}\)

(a) determine the domains of \(f\) and \(g,\) (b) find any vertical asymptotes and holes in the graphs of \(f\) and \(g,\) (c) compare \(f\) and \(g\) by completing the table, (d) use a graphing utility to graph \(f\) and \(g,\) and (e) explain why the differences in the domains of \(f\) and \(g\) are not shown in their graphs. $$f(x)=\frac{x^{2}-16}{x-4}, \quad g(x)=x+4$$ $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline f(x) & & & & & & & \\ \hline g(x) & & & & & & & \\ \hline \end{array}$$

Find (a) \(f \circ g\) and (b) \(g \circ f\). $$f(x)=x^{3}-1, g(x)=\sqrt[3]{x+1}$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{x^{2}+3 x}{x^{2}+x-6}$$

Use synthetic division to divide. $$\left(x^{3}+512\right) \div(x+8)$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=3 x^{3}-5 x^{2}+48 x-80$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(h(x)=4 x^{2}-4 x+21\)

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(f(x)=2 x^{4}-3 x+1\)

(a) determine the domains of \(f\) and \(g,\) (b) find any vertical asymptotes and holes in the graphs of \(f\) and \(g,\) (c) compare \(f\) and \(g\) by completing the table, (d) use a graphing utility to graph \(f\) and \(g,\) and (e) explain why the differences in the domains of \(f\) and \(g\) are not shown in their graphs. $$f(x)=\frac{x^{2}-9}{x-3}, \quad g(x)=x+3$$ $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline f(x) & & & & & & & \\ \hline g(x) & & & & & & & \\ \hline \end{array}$$

Find (a) \(f \circ g\) and (b) \(g \circ f\). $$f(x)=\sqrt[3]{x+5}, g(x)=x^{3}-5$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$g(x)=\frac{5(x+4)}{x^{2}+x-12}$$

Use synthetic division to divide. $$\left(x^{3}-729\right) \div(x-9)$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=3 x^{3}-2 x^{2}+75 x-50$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=2 x^{2}-x+1\)

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(h(x)=1-x^{6}\)

(a) determine the domains of \(f\) and \(g,\) (b) find any vertical asymptotes and holes in the graphs of \(f\) and \(g,\) (c) compare \(f\) and \(g\) by completing the table, (d) use a graphing utility to graph \(f\) and \(g,\) and (e) explain why the differences in the domains of \(f\) and \(g\) are not shown in their graphs. $$f(x)=\frac{x^{2}-1}{x^{2}-2 x-3}, \quad g(x)=\frac{x-1}{x-3}$$ $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & & & & & & & \\ \hline g(x) & & & & & & & \\ \hline \end{array}$$

Determine algebraically whether the function is one-to-one. If it is, find its inverse function. Verify your answer graphically. $$f(x)=2 x+5$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{x^{2}-1}{x+1}$$

Use synthetic division to divide. $$\frac{4 x^{3}+16 x^{2}-23 x-15}{x+\frac{1}{2}}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(t)=t^{3}-3 t^{2}-15 t+125$$

Describe the graph of the quadratic function. Identify the vertex and \(x\) -intercept(s). Use a graphing utility to verify your results. \(g(x)=x^{2}+8 x+11\)

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(g(x)=5-\frac{7}{2} x-3 x^{2}\)

(a) determine the domains of \(f\) and \(g,\) (b) find any vertical asymptotes and holes in the graphs of \(f\) and \(g,\) (c) compare \(f\) and \(g\) by completing the table, (d) use a graphing utility to graph \(f\) and \(g,\) and (e) explain why the differences in the domains of \(f\) and \(g\) are not shown in their graphs. $$f(x)=\frac{x^{2}-4}{x^{2}-3 x+2}, \quad g(x)=\frac{x+2}{x-1}$$ $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline f(x) & & & & & & & \\ \hline g(x) & & & & & & & \\ \hline \end{array}$$

Determine algebraically whether the function is one-to-one. If it is, find its inverse function. Verify your answer graphically. $$f(x)=\frac{x-4}{5}$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{x^{2}-16}{x-4}$$

Use synthetic division to divide. $$\frac{3 x^{3}-4 x^{2}+5}{x-\frac{3}{2}}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=x^{3}+11 x^{2}+39 x+29$$

Describe the graph of the quadratic function. Identify the vertex and \(x\) -intercept(s). Use a graphing utility to verify your results. \(f(x)=x^{2}+10 x+14\)

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(f(x)=\frac{1}{3} x^{3}+5 x\)

Determine algebraically whether the function is one-to-one. If it is, find its inverse function. Verify your answer graphically. $$f(x)=x^{2}+5, x \geq 0$$

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$f(x)=\frac{2+x}{1-x}$$

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to verify that the expressions are equivalent. Verify the results algebraically. $$y_{1}=\frac{x^{2}}{x+2}, \quad y_{2}=x-2+\frac{4}{x+2}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(x)=5 x^{3}-9 x^{2}+28 x+6$$

Describe the graph of the quadratic function. Identify the vertex and \(x\) -intercept(s). Use a graphing utility to verify your results. \(f(x)=-\left(x^{2}-2 x-15\right)\)

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(f(x)=\frac{6 x^{5}-2 x^{4}+4 x^{2}-5 x}{3}\)

Determine algebraically whether the function is one-to-one. If it is, find its inverse function. Verify your answer graphically. $$f(x)=2 x^{2}-3, x \geq 0$$

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$f(x)=\frac{3-x}{2-x}$$

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to verify that the expressions are equivalent. Verify the results algebraically. $$y_{1}=\frac{x^{2}+2 x-1}{x+3}, \quad y_{2}=x-1+\frac{2}{x+3}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$f(s)=3 s^{3}-4 s^{2}+8 s+8$$

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(f(x)=\frac{3 x^{7}-2 x^{5}+5 x^{3}+6 x^{2}}{4}\)

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$1-3 i$$

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$g(x)=\frac{3 x-4}{-x}$$