Chapter 4

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$\sqrt{x} \cdot \sqrt[3]{x}$$

Find all complex solutions for each equation by hand. Do not use a calculator. $$\frac{x}{x-3}+\frac{4}{x+3}=\frac{18}{x^{2}-9}$$

Solve each equation by hand. Do not use a calculator. $$5 x^{-2}+13 x^{-1}=28$$

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}}\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=-\frac{2}{x^{2}}$$

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$(\sqrt[5]{z})^{-3}$$

Find all complex solutions for each equation by hand. Do not use a calculator. $$\frac{2 x}{x-3}+\frac{4}{x+3}=\frac{24}{9-x^{2}}$$

Solve each equation by hand. Do not use a calculator. $$3 x^{-2}-19 x^{-1}+20=0$$

Explain why the graph of the rational function \(f(x)=\frac{-1}{x^{2}+4}\) has no vertical asymptotes.

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}}\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=\frac{0.5}{x^{2}}$$

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$\sqrt{y \cdot \sqrt{y}}$$

Find all complex solutions for each equation by hand. Do not use a calculator. $$9 x^{-1}+4(6 x-3)^{-1}=2(6 x-3)^{-1}$$

Solve each equation by hand. Do not use a calculator. $$x^{2 / 3}-x^{1 / 3}-6=0$$

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x+1}{x-4}$$

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}}\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=\frac{1}{(x-3)^{2}}$$

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$\frac{\sqrt[3]{x}}{\sqrt{x}}$$

Find all complex solutions for each equation by hand. Do not use a calculator. $$x(x-2)^{-1}+x(x+2)^{-1}=2\left(x^{2}-4\right)^{-1}$$

Solve each equation by hand. Do not use a calculator. $$x^{2 / 3}+9 x^{1 / 3}+14=0$$

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x-5}{x+3}$$

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}}\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=\frac{-2}{(x-3)^{2}}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[3]{-4}$$

Solve each equation by hand. Do not use a calculator. $$x^{3 / 4}-x^{1 / 2}-x^{1 / 4}+1=0$$

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x+2}{x-3}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[5]{-3}$$

Solve each equation by hand. Do not use a calculator. $$x^{3 / 4}-2 x^{1 / 2}-4 x^{1 / 4}+8=0$$

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x-3}{x+4}$$

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}}\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=\frac{-1}{(x-4)^{2}}+2$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[3]{-125}$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt{3 x+7}=2\) (b) \(\sqrt{3 x+7}>2\) (c) \(\sqrt{3 x+7}<2\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{4-2 x}{8-x}$$

Use the methods of Examples 1 and 3 to solve the rational equation and associated inequalities given.Then, support your answer by using the \(x\) -intercept method with a calculator graph in the suggested window. (a) \(\frac{x-6}{x+2}=-1\) (b) \(\frac{x-6}{x+2}<-1\) (c) \(\frac{x-6}{x+2}>-1\) Window: \([-10,10]\) by \([-10,10]\)

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[5]{-243}$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt{2 x+13}=3\) (b) \(\sqrt{2 x+13}>3\) (c) \(\sqrt{2 x+13}<3\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{6-3 x}{4-x}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[3]{-17}$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt{4 x+13}=2 x-1\) (b) \(\sqrt{4 x+13}>2 x-1\) (c) \(\sqrt{4 x+13}<2 x-1\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{3 x}{(x+1)(x-2)}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[5]{-8}$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt{3 x+7}=3 x+5\) (b) \(\sqrt{3 x+7}>3 x+5\) (c) \(\sqrt{3 x+7}<3 x+5\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{2 x+1}{(x+2)(x+4)}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[6]{\pi^{2}}$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt{5 x+1}+2=2 x\) (b) \(\sqrt{5 x+1}+2>2 x\) (c) \(\sqrt{5 x+1}+2<2 x\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{5 x}{x^{2}-1}$$

Suppose that the graph of a rational function \(f\) has vertical asymptote \(x=1\) horizontal asymptote \(y=2,\) domain ( \(-\infty, 1\) ) U ( \(1, \infty\) ), and range ( \(-\infty, 2\) ) U ( \(2, \infty\) ). Give the vertical asymptote, horizontal asymptote, domain, and range for the graph of each shifted function. $$y=f(x+2)-1$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[6]{\pi^{-1}}$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt{3 x+4}+x=8\) (b) \(\sqrt{3 x+4}+x>8\) (c) \(\sqrt{3 x+4}+x<8\)

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x}{4-x^{2}}$$

Use a calculator to find each root or power. Give as many digits as your display shows. $$13^{-1 / 3}$$

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(x+6)(x-2)}{(x+3)(x-4)}$$

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt{3 x-6}+2=\sqrt{5 x-6}\) (b) \(\sqrt{3 x-6}+2>\sqrt{5 x-6}\) (c) \(\sqrt{3 x-6}+2<\sqrt{5 x-6}\)

Use a calculator to find each root or power. Give as many digits as your display shows. $$15^{-1 / 6}$$