Chapter 5

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

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

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) \(\sqrt[3]{x^{2}-2 x}=\sqrt[3]{x}\) (b) \(\sqrt[3]{x^{2}-2 x}>\sqrt[3]{x}\) (c) \(\sqrt[3]{x^{2}-2 x}<\sqrt[3]{x}\)

Use a calculator to find each root or power. Give as many digits as your display shows. $$32^{0.2}$$

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

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) \(\sqrt[3]{4 x^{2}-4 x+1}=\sqrt[3]{x}\) (b) \(\sqrt[3]{4 x^{2}-4 x+1}>\sqrt[3]{x}\) (c) \(\sqrt[3]{4 x^{2}-4 x+1}<\sqrt[3]{x}\)

Use a calculator to find each root or power. Give as many digits as your display shows. $$81^{0.25}$$

Solve each equation and inequality. (These types of equations and inequalities occur in calculus.) (a) \(\frac{(x-2)(2)-(2 x+1)(1)}{(x-2)^{2}}=0\) (b) \(\frac{(x-2)(2)-(2 x+1)(1)}{(x-2)^{2}}<0\)

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

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) \(\sqrt[4]{3 x+1}=1\) (b) \(\sqrt[4]{3 x+1}>1\) (c) \(\sqrt[4]{3 x+1}<1\)

Use a calculator to find each root or power. Give as many digits as your display shows. $$\left(\frac{5}{6}\right)^{-1.3}$$

Solve each equation and inequality. (These types of equations and inequalities occur in calculus.) (a) \(\frac{\left(x^{2}-1\right)(1)-(x+1)(2 x)}{\left(x^{2}-1\right)^{2}}=0\) (b) \(\frac{\left(x^{2}-1\right)(1)-(x+1)(2 x)}{\left(x^{2}-1\right)^{2}}>0\)

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

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) \(\sqrt[4]{x-15}=2\) (b) \(\sqrt[4]{x-15}>2\) (c) \(\sqrt[4]{x-15}<2\)

Use a calculator to find each root or power. Give as many digits as your display shows. $$\left(\frac{4}{7}\right)^{-0,6}$$

Solve each equation and inequality. (These types of equations and inequalities occur in calculus.) (a) \(\frac{\left(x^{2}+1\right)(2 x)-\left(x^{2}-1\right)(2 x)}{\left(x^{2}+1\right)^{2}}=0\) (b) \(\frac{\left(x^{2}+1\right)(2 x)-\left(x^{2}-1\right)(2 x)}{\left(x^{2}+1\right)^{2}} \geq 0\)

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

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) \((2 x-5)^{1 / 2}-2=(x-2)^{1 / 2}\) (b) \((2 x-5)^{1 / 2}-2 \geq(x-2)^{1 / 2}\) (c) \((2 x-5)^{1 / 2}-2 \leq(x-2)^{1 / 2}\)

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

Solve each equation and inequality. (These types of equations and inequalities occur in calculus.) (a) \(\frac{\left(x^{2}-1\right)(3)-(3 x-1)(2 x)}{\left(x^{2}-1\right)^{2}}=0\) (b) \(\frac{\left(x^{2}-1\right)(3)-(3 x-1)(2 x)}{\left(x^{2}-1\right)^{2}} \leq 0\)

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

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) \((x+5)^{1 / 2}-2=(x-1)^{1 / 2}\) (b) \((x+5)^{1 / 2}-2 \geq(x-1)^{1 / 2}\) (c) \((x+5)^{1 / 2}-2 \leq(x-1)^{1 / 2}\)

Use a calculator to find each root or power. Give as many digits as your display shows. $$(2 \pi)^{43}$$

Solve each equation and inequality. (These types of equations and inequalities occur in calculus.) (a) \(\frac{(2 x+1)(2 x)-\left(x^{2}+1\right)(2)}{(2 x+1)^{2}}=0\) (b) \(\frac{(2 x+1)(2 x)-\left(x^{2}+1\right)(2)}{(2 x+1)^{2}}<0\)

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

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) \(\left(x^{2}+6 x\right)^{1 / 4}=2\) (b) \(\left(x^{2}+6 x\right)^{1 / 4}>2\) (c) \(\left(x^{2}+6 x\right)^{1 / 4}<2\)

Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{1.62}, \quad x=1.2$$

Solve each equation and inequality. (These types of equations and inequalities occur in calculus.) (a) \(\frac{(x-1)(2 x)-\left(x^{2}\right)(1)}{(x-1)^{2}}=0\) (b) \(\frac{(x-1)(2 x)-\left(x^{2}\right)(1)}{(x-1)^{2}}>0\)

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

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) \(\left(x^{2}+2 x\right)^{1 / 4}=3^{1 / 4}\) (b) \(\left(x^{2}+2 x\right)^{1 / 4}>3^{1 / 4}\) (c) \(\left(x^{2}+2 x\right)^{1 / 4}<3^{1 / 4}\)

Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{-0.71}, \quad x=3.8$$

CONCEPT CHECK In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$ \frac{1}{x^{2}+1}>0 $$ The numerator of the rational expression, 1, is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nonnegative number, \(x^{2},\) and a positive number, 1. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than \(0,\) and this will always be true, the solution set is \((-\infty, \infty)\) Use similar reasoning to solve each inequality. $$\frac{-1}{x^{2}+2}<0$$

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

Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{3 / 2}-x^{1 / 2}, \quad x=50$$

CONCEPT CHECK In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$ \frac{1}{x^{2}+1}>0 $$ The numerator of the rational expression, 1, is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nonnegative number, \(x^{2},\) and a positive number, 1. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than \(0,\) and this will always be true, the solution set is \((-\infty, \infty)\) Use similar reasoning to solve each inequality. $$\frac{5}{x^{2}+2}<0$$

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

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) \((x-3)^{2 / 5}=(4 x)^{1 / 5}\) (b) \((x-3)^{2 / 5}>(4 x)^{1 / 5}\) (c) \((x-3)^{2 / 5}<(4 x)^{1 / 5}\)

Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{5 / 4}-x^{-3 / 4}, \quad x=7$$

CONCEPT CHECK In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$ \frac{1}{x^{2}+1}>0 $$ The numerator of the rational expression, 1, is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nonnegative number, \(x^{2},\) and a positive number, 1. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than \(0,\) and this will always be true, the solution set is \((-\infty, \infty)\) Use similar reasoning to solve each inequality. $$\frac{-5}{x^{2}+2}>0$$

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

Use analytic or graphical methods to solve the inequality. $$\sqrt{2 x+3}>3$$

CONCEPT CHECK In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$ \frac{1}{x^{2}+1}>0 $$ The numerator of the rational expression, 1, is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nonnegative number, \(x^{2},\) and a positive number, 1. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than \(0,\) and this will always be true, the solution set is \((-\infty, \infty)\) Use similar reasoning to solve each inequality. $$\frac{x^{4}+2}{-6} \leq 0$$

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

Use analytic or graphical methods to solve the inequality. $$\sqrt{3 x-5} \leq 4$$

CONCEPT CHECK In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$ \frac{1}{x^{2}+1}>0 $$ The numerator of the rational expression, 1, is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nonnegative number, \(x^{2},\) and a positive number, 1. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than \(0,\) and this will always be true, the solution set is \((-\infty, \infty)\) Use similar reasoning to solve each inequality. $$\frac{x^{4}+2}{-6} \geq 0$$

Sketch a graph of 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 analytic or graphical methods to solve the inequality. $$\sqrt{2 x-2}-1<\sqrt{x}$$

CONCEPT CHECK In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$ \frac{1}{x^{2}+1}>0 $$ The numerator of the rational expression, 1, is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nonnegative number, \(x^{2},\) and a positive number, 1. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than \(0,\) and this will always be true, the solution set is \((-\infty, \infty)\) Use similar reasoning to solve each inequality. $$\frac{x^{4}+x^{2}+3}{x^{2}+2}<0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(x+3)(x-5)}{(x+1)(x-4)}$$

Use analytic or graphical methods to solve the inequality. $$\sqrt{4 x}-1 \geq \sqrt{2 x+1}$$