Chapter 5

Evaluate each expression. $$\sqrt{169}$$

In Exercises begin by drawing a rough sketch to determine the number of real solutions for the equation \(y_{1}=y_{2}\). Then solve this equation by hand. Give the solution set and any extraneous values that may occur. Do not use a calculator. $$\begin{aligned} &y_{1}=\sqrt{x}\\\ &y_{2}=2 x-1 \end{aligned}$$

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. \(\mathbf{I}\) \(f(x)=\frac{x+7}{x+1}\) \(\mathbf{II}\) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote.

Provide a short answer to each question. Do not use a calculator. What is the domain of \(f(x)=\frac{1}{x} ?\) What is its range?

Evaluate each expression. $$-\sqrt[3]{64}$$

In Exercises begin by drawing a rough sketch to determine the number of real solutions for the equation \(y_{1}=y_{2}\) Then solve this equation by hand. Give the solution set and any extraneous values that may occur. Do not use a calculator. $$\begin{aligned} &y_{1}=\sqrt{x}\\\ &y_{2}=x-6 \end{aligned}$$

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. \(\mathbf{I}\) \(f(x)=\frac{x+10}{x+2}\) \(\mathbf{II}\) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote.

Provide a short answer to each question. Do not use a calculator. What is the domain of \(f(x)=\frac{1}{x^{2}}\) ? What is its range?

Evaluate each expression. $$\sqrt[5]{-32}$$

In Exercises begin by drawing a rough sketch to determine the number of real solutions for the equation \(y_{1}=y_{2}\) Then solve this equation by hand. Give the solution set and any extraneous values that may occur. Do not use a calculator. $$\begin{aligned} &y_{1}=\sqrt{x}\\\ &y_{2}=-x+3 \end{aligned}$$

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. \(\mathbf{I}\) \(f(x)=\frac{1}{x+12}\) \(\mathbf{II}\) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote.

Provide a short answer to each question. Do not use a calculator. What is the largest open interval over which \(f(x)=\frac{1}{x}\) increases? decreases? is constant?

In Exercises begin by drawing a rough sketch to determine the number of real solutions for the equation \(y_{1}=y_{2}\) Then solve this equation by hand. Give the solution set and any extraneous values that may occur. Do not use a calculator. $$\begin{aligned} &y_{1}=\sqrt{x}\\\ &y_{2}=3 x \end{aligned}$$

Evaluate each expression. $$\sqrt[4]{16}$$

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. \(\mathbf{I}\) \(f(x)=\frac{-3+x^{2}}{x^{2}}\) \(\mathbf{II}\) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote.

In Exercises begin by drawing a rough sketch to determine the number of real solutions for the equation \(y_{1}=y_{2}\) Then solve this equation by hand. Give the solution set and any extraneous values that may occur. Do not use a calculator. $$\begin{aligned} &y_{1}=\sqrt[3]{x}\\\ &y_{2}=x^{2} \end{aligned}$$

Evaluate each expression. $$81^{3 / 2}$$

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. \(\mathbf{I}\) \(f(x)=\frac{x^{2}-16}{x+4}\) \(\mathbf{II}\) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote.

Provide a short answer to each question. Do not use a calculator. What is the equation of the vertical asymptote of the graph of \(y=\frac{1}{x-3}+2 ?\) of the horizontal asymptote?

Use a hand-drawn graph to explain why \(\sqrt{x}=-x-5\) has no real solutions.

Evaluate each expression. $$27^{4 / 3}$$

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. \(\mathbf{I}\) \(f(x)=\frac{4 x+3}{x-7}\) \(\mathbf{II}\) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote.

Provide a short answer to each question. Do not use a calculator. What is the equation of the vertical asymptote of the graph of \(y=\frac{1}{(x+2)^{2}}-4 ? \quad\) of the horizontal asymptote?

Check that proposed solutions \(\frac{3}{2}\) and \(\frac{5}{3}\) from Example 6 are solutions of \(15 x^{-2}-19 x^{-1}+6=0\).

Evaluate each expression. $$125^{-2 / 3}$$

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. \(\mathbf{I}\) \(f(x)=\frac{x^{2}+3 x+4}{x-5}\) \(\mathbf{II}\) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote.

Provide a short answer to each question. Do not use a calculator. Is \(f(x)=\frac{1}{x^{2}}\) an even or odd function? What symmetry does its graph exhibit?

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. \(\mathbf{I}\) \(f(x)=\frac{x+3}{x-6}\) \(\mathbf{II}\) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote.

Evaluate each expression. $$(\sqrt[3]{-27})^{2}$$

Provide a short answer to each question. Do not use a calculator. Is \(f(x)=\frac{1}{x}\) an even or odd function? What symmetry does its graph exhibit?

Solve each equation by hand. Do not use a calculator. $$x-4=\sqrt{3 x-8}$$

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. State the domain of \(f .\) $$f(x)=\frac{3}{x-5}$$

Evaluate each expression. $$(-1000)^{2 / 3}$$

Solve each equation by hand. Do not use a calculator. $$x-5=\sqrt{5 x-1}$$

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. State the domain of \(f .\) $$f(x)=\frac{-6}{x+9}$$

Evaluate each expression. $$(-125)^{-4 / 3}$$

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. State the domain of \(f .\) $$f(x)=\frac{4-3 x}{2 x+1}$$

Solve each equation by hand. Do not use a calculator. $$\sqrt{x+5}+1=x$$

Evaluate each expression. $$8^{2 / 3}$$

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. State the domain of \(f .\) $$f(x)=\frac{2 x+6}{x-4}$$

Solve each equation by hand. Do not use a calculator. $$\sqrt{4-3 x}-8=x$$

Find all complex solutions for each equation by hand. $$\frac{2 x}{x^{2}-1}=\frac{2}{x+1}-\frac{1}{x-1}$$

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. State the domain of \(f .\) $$f(x)=\frac{x^{2}-1}{x+3}$$

Solve each equation by hand. Do not use a calculator. $$\sqrt{2 x+3}-\sqrt{x+1}=1$$

Evaluate each expression. $$16^{-3 / 4}$$

Find all complex solutions for each equation by hand. $$\frac{8 x}{4 x^{2}-1}=\frac{3}{2 x+1}+\frac{3}{2 x-1}$$

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. State the domain of \(f .\) $$f(x)=\frac{x^{2}+4}{x-1}$$

Solve each equation by hand. Do not use a calculator. $$\sqrt{3 x+4}-\sqrt{2 x-4}=2$$

Evaluate each expression. $$25^{-3 / 2}$$

Find all complex solutions for each equation by hand. $$\frac{4}{x^{2}-3 x}-\frac{1}{x^{2}-9}=0$$