Chapter 7

Solve each equation. $$ 125 x^{3}-1=0 $$

Absolute value symbols are needed when you simplify some of these expressions. Which ones are they? $$\text { I. }\sqrt[3]{-x^{3} y^{6}} \quad \text { II. } \sqrt{x^{2} y^{4}} \quad \text { III. } \sqrt[4]{x^{8} y^{12}} \quad \text { IV. } \sqrt{x^{4} y^{6}}$$ F. II and III only H. II, III and IV only G. II and IV only J. II, III and IV

Simplify each expression. Assume that all variables are positive. $$\left(x^{\frac{1}{2}} \cdot x^{\frac{5}{12}}\right)^{\frac{1}{3}} \div x^{\frac{2}{3}}$$

Which equation shows \(y+3=\sqrt{\frac{x}{16}+2}\) rewritten in the form \(y=a \sqrt{x-h}+k ?\) F. \(y=\frac{3}{4} \sqrt{x-(-2)}\) G. \(y=\frac{1}{4} \sqrt{x-(-2)}+(-3)\) H. \(y=\frac{1}{4} \sqrt{x-(-32)}+(-3)\) J. \(y=\frac{1}{8} \sqrt{x+32}+(-3)\)

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ (f \circ g)(x) $$

Simplify each expression. \(100^{-\frac{3}{2}} \cdot 0.01^{\frac{3}{2}}\)

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ (f-g)(x) $$

For what values of \(x\) is \(\sqrt{x} \cdot \sqrt{-x}\) a real number? Explain.

Solve each equation. $$ x^{4}-14 x^{2}+49=0 $$

For what values of \(x\) and \(y\) does \(\sqrt{x^{2} y^{4}}\) equal \(\sqrt[3]{x^{3}} y^{6} ?\) Explain your answer.

Simplify each expression. Assume that all variables are positive. $$\left(x^{\frac{3}{4}} \div x^{\frac{7}{8}}\right) \cdot x^{-\frac{1}{6}}$$

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ (g \circ f)(x) $$

Evaluate each expression. \(_{7} \mathrm{P}_{1}\)

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ (f \circ g)(x) $$

Rationalize the denominator of \(\sqrt[3]{\frac{3}{2 x}}\) . Explain your steps.

Solve each equation. $$ 25 x^{4}-40 x^{2}+16=0 $$

Expand each binomial. $$ (x+y)^{5} $$

Simplify each expression. Assume that all variables are positive. $$\left[\left(x^{-\frac{1}{2}}\right)^{2}\right]^{\frac{1}{3}}$$

Compare the domains and ranges of the functions \(f(x)=\sqrt{x-1}\) and \(g(x)=\sqrt{x}-1\)

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ (h \circ g)(x) $$

Evaluate each expression. \(_{7} \mathrm{P}_{3}\)

Grades Suppose your teacher offers to give the whole class a bonus if everyone passes the next math test. The teacher says she will \((1)\) give everyone a 10 -point bonus and \((2)\) increase everyone's grade by 9\(\%\) of their score. a. Let \(x\) represent the original test scores. Write statements \((1)\) and \((2)\) as the functions \(f(x)\) and \(g(x),\) respectively. b. Explain the meaning of \(f(g(x))\) . Evaluate \(f(g(75))\) . c. Explain the meaning of \(g(f(x)) .\) Evaluate \(g(f(75))\) . d. Does \(g(f(x))=f(g(x)) ?\)

Simplify each radical expression. Use absolute value symbols as needed. $$ \sqrt{121 a^{90}} $$

Solve each equation. $$ 81 x^{4}-1=0 $$

Expand each binomial. $$ (2-3 y)^{4} $$

Simplify each expression. Assume that all variables are positive. $$\left[\left(\sqrt{x^{3} y^{3}}\right)^{\frac{1}{3}}\right]^{-1}$$

Find the inverse of each function. Is the inverse a function? \(f(x)=4 x-1\)

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ g(x)+g(x) $$

Evaluate each expression. \(_{5} \mathrm{P}_{3}\)

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ f(x) \cdot g(x) $$

Simplify each radical expression. Use absolute value symbols as needed. $$ -\sqrt{81 c^{48} d^{64}} $$

Expand each binomial. $$ (3 x-5)^{6} $$

Writing Explain why \((-64)^{\frac{1}{3}}=-64^{\frac{1}{3}}\) and \((-64)^{\frac{1}{2}} \neq-64^{\frac{1}{2}}\).

Find the inverse of each function. Is the inverse a function? \(f(x)=\frac{2}{3} x-3\)

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ (h \circ(g \circ f))(x) $$

Evaluate each expression. \(_{8} P_{4}\)

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ \frac{f(x)}{g(x)} $$

Simplify each radical expression. Use absolute value symbols as needed. $$ \sqrt[3]{-64 a^{81}} $$

Expand each binomial. $$ (2 a-b)^{7} $$

Error Analysis Explain why the following simplification is incorrect. \(5\left(4-5^{\frac{1}{2}}\right)=5(4)-5\left(5^{\frac{1}{2}}\right)=20-25^{\frac{1}{2}}=15\)

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ (f \circ g)(x)+h(x) $$

Evaluate each expression. \(_{4} \mathrm{P}_{4}\)

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ \frac{g(x)}{f(x)} $$

Simplify each radical expression. Use absolute value symbols as needed. $$ \sqrt[5]{32 y^{25}} $$

Write each function in factored form. Check by multiplying. $$ y=4 x^{3}-49 x $$

Open-Ended Find three nonzero numbers \(a\) such that \(a\left(4+5^{\frac{1}{2}}\right)\) is a rational number. Can \(a\) itself be a rational number? Explain.

Find the inverse of each function. Is the inverse a function? \(f(x)=\sqrt{x+3}-4\)

Find each indicated root if it is a real number. $$ \sqrt[4]{16} $$

Evaluate each expression. \(_{5} \mathrm{C}_{2}\)

Find each composition of functions. Simplify your answer. Let \(f(x)=\frac{1}{x} .\) Find \(f(f(x))\)