Chapter 7

Graph. Find the domain and the range of each function. \(y=-3 \sqrt{x}+2\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=(x+1)^{2}-1 $$

Solve. Check for extraneous solutions. \(2(x-1)^{\frac{4}{3}}+4=36\)

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (g \circ f)(c) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \frac{4+\sqrt{6}}{\sqrt{2}+\sqrt{3}} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ 5 \sqrt{2 x y^{6}} \cdot 2 \sqrt{2 x^{3} y} $$

Write each expression in simplest form. Assume that all variables are positive. $$\left(3 x^{\frac{2}{3}}\right)^{-1}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[3]{\frac{8}{216}} $$

Graph. Find the domain and the range of each function. \(y=-\frac{4}{5} \sqrt{x}\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=x^{3} $$

Solve. Check for extraneous solution. $$x^{\frac{1}{2}}-(x-5)^{\frac{1}{2}}=2$$

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (f \circ g)(-a) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \frac{5-\sqrt{21}}{\sqrt{3}-\sqrt{7}} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt{2}(\sqrt{50}+7) $$

Write each expression in simplest form. Assume that all variables are positive. $$5\left(x^{\frac{2}{3}}\right)^{-1}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[4]{0.0016} $$

Graph. Find the domain and the range of each function. \(y=7-\sqrt{2 x-1}\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=x^{4} $$

Solve. Check for extraneous solutions. \(\sqrt{x}=\sqrt{x-8}+2\)

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (g \circ f)(-a) $$

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \frac{3+\sqrt[3]{2}}{\sqrt[3]{2}} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ 3(5+\sqrt{21}) $$

Write each expression in simplest form. Assume that all variables are positive. $$\left(-27 x^{-9}\right)^{\frac{1}{3}}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[4]{\frac{1}{256}} $$

Graph. Find the domain and the range of each function. \(y=4 \sqrt[3]{x-2}+1\)

Find the inverse of each function. Is the inverse a function? $$ f(x)=\frac{2 x^{2}}{5}+1 $$

Sales A car dealer offers a 10\(\%\) discount off the list price \(x\) for any car on the lot. At the same time, the manufacturer offers a \(\$ 2000\) rebate for each purchase of a car. a. Write a function \(f(x)\) to represent the price after the discount. b. Write a function \(g(x)\) to represent the price after the \(\$ 2000\) rebate. c. Suppose the list price of a car is \(\$ 18,000\) . Use a composite function to find the price of the car if the discount is applied before the rebate. d. Suppose the list price of a car is \(\$ 18,000\) . Use a composite function to find the price of the car if the rebate is applied before the discount.

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \frac{5+\sqrt[4]{x}}{\sqrt[4]{x}} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt{5}(\sqrt{5}+\sqrt{15}) $$

Write each expression in simplest form. Assume that all variables are positive. $$\left(-32 y^{15}\right)^{\frac{1}{3}}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[4]{16 c^{4}} $$

Graph. Find the domain and the range of each function. \(y=\frac{1}{2} \sqrt{x-1}+3\)

Water Supply The velocity of the water that flows from an opening at the base of a tank depends on the height of water above the opening. The function \(v(x)=\sqrt{2 g x}\) models the velocity \(v\) in feet per second where \(g\) , the acceleration due to gravity, is about 32 \(\mathrm{ft} / \mathrm{s}^{2}\) and \(x\) is the height in feet of the water. Find the inverse function and use it to find the depth of water when the flow is 40 \(\mathrm{ft} / \mathrm{s}\) , and when the flow is 20 \(\mathrm{ft} / \mathrm{s}\) .

Economics Suppose the function \(f(x)=0.12 x\) represents the number of U.S. dollars equivalent to \(x\) Chinese yuan and the function \(g(x)=9.14 x\) represents the number of Mexican pesos equivalent to \(x\) U.S. dollars. a. Write a composite function that represents the number of Mexican pesos equivalent to \(x\) Chinese yuan. b. Find the value in Mexican pesos of an item that costs 15 Chinese yuan.

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \frac{4-2 \sqrt[3]{6}}{\sqrt[3]{4}} $$

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{2 x} \cdot \sqrt[3]{4} \cdot \sqrt[3]{2 x^{2}} $$

Write each expression in simplest form. Assume that all variables are positive. $$\left(\frac{x^{3}}{x^{-1}}\right)^{-\frac{1}{4}}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[3]{81 x^{3} y^{6}} $$

Graph. Find the domain and the range of each function. \(y=-3 \sqrt[3]{x-4}-3\)

Writing Explain how you can find the range of the inverse of \(f(x)=\sqrt{x-1}\) without finding the inverse itself.

Let \(f(x)=2 x+5\) and \(g(x)=x^{2}-3 x+2 .\) Perform each function operation. $$ f(x)+g(x) $$

The golden ratio is \(\frac{1+\sqrt{5}}{2} .\) Find the difference between the golden ratio and its reciprocal.

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{3 x^{2}} \cdot \sqrt[3]{x^{2}} \cdot \sqrt[3]{9 x^{3}} $$

Write each expression in simplest form. Assume that all variables are positive. $$\left(\frac{x^{2}}{x^{-11}}\right)^{\frac{1}{3}}$$

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt{144 x^{3} y^{4} z^{5}} $$

Graph. Find the domain and the range of each function. \(y=-\sqrt{x+\frac{1}{2}}\)

A function consists of the pairs \((2,3),(x, 4)\) and \((5,6) .\) What values, if any, may \(x\) not assume?

Let \(f(x)=2 x+5\) and \(g(x)=x^{2}-3 x+2 .\) Perform each function operation. $$ 3 f(x)-2 $$

Critical Thinking Describe the possible values of \(a\) such that \(\sqrt{72}+\sqrt{a}\) can be simplified to a single term.

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{\sqrt{5 x^{4}}}{\sqrt{2 x^{2} y^{3}}} $$