Chapter 5

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=\frac{4 x}{x+3} $$

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{1}{2 x} $$

Sketch a graph of \(y=f(x)\) $$ f(x)=\left(\frac{1}{4}\right)^{x} $$

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=\frac{x+3}{2} $$

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{2}{\sqrt{x}} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. (a) \(10^{x}=0.01 (b) \)10^{x}=7\( (c) \)10^{x}=-4$

Exercises \(53-56:\) Use the given \(f(x)\) and \(g(x)\) to evaluate each expression. \(f(x)=\sqrt{x+5}, \quad g(x)=x^{2}\) (a) \((f \circ g)(2)\) (b) \((g \circ f)(-1)\)

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{1}{2}(4-5 x)+1 $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. (a) \(10^{x}=1000 (b) \)10^{x}=5\( (c) \)10^{x}=-2$

Exercises \(53-56:\) Use the given \(f(x)\) and \(g(x)\) to evaluate each expression. $$ \begin{aligned} &f(x)=\left|x^{2}-4\right|, \quad g(x)=2 x^{2}+x+1\\\ &\begin{array}{lll} \text { (a) }(f \circ g)(1) & \text { (b) }(g \circ f)(-3) \end{array} \end{aligned} $$

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=6-\frac{3}{4}(2 x-4) $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. (a) \(4^{x}=\frac{1}{16}\) (b) \(e^{x}=2\) (c) \(5^{x}=125\)

Exercises \(53-56:\) Use the given \(f(x)\) and \(g(x)\) to evaluate each expression. \(f(x)=5 x-2, g(x)=|x|\) (a) \((f \circ g)(-4)\) (b) \((g \circ f)(5)\)

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{x}{x+2} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. (a) \(2^{x}=9\) (b) \(10^{x}=\frac{1}{1000}\) (c) \(e^{x}=8\)

Exercises \(53-56:\) Use the given \(f(x)\) and \(g(x)\) to evaluate each expression. \(f(x)=\frac{1}{x-4}, g(x)=5\) (a) \((f \circ g)(3)\) (b) \((g \circ f)(8)\)

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{3 x}{x-1} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. (a) \(9^{x}=1\) (b) \(10^{x}=\sqrt{10}\) (c) \(4^{x}=\sqrt[3]{4}\)

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=x^{3}, \quad g(x)=x^{2}+3 x-1 $$

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{2 x+1}{x-1} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. (a) \(2^{x}=\sqrt{8}\) (b) \(7^{x}=1\) (c) \(e^{x}=\sqrt[y]{e}\)

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=2-x, \quad g(x)=\frac{1}{x^{2}} $$

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{1-x}{3 x+1} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$e^{-x}=3$$

Complete the following. A.Use a table of \(f(x)\) and \(g(x)\) to determine whether \(f(x)=g(x) B.If possible, use properties of logarithms to show that \)f(x)=g(x)$ $$ f(x)=\ln 2 x^{2}-\ln x, \quad g(x)=\ln 2 x $$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=x+2, \quad g(x)=x^{4}+x^{2}-3 x-4 $$

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{1}{x}-3 $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$e^{-x}=\frac{1}{2}$$

Complete the following. A.Use a table of \(f(x)\) and \(g(x)\) to determine whether \(f(x)=g(x) B.If possible, use properties of logarithms to show that \)f(x)=g(x)$ $$ f(x)=\log x^{2}+\log x^{3}, \quad g(x)=5 \log x $$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=x^{2}, \quad g(x)=\sqrt{1-x} $$

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{1}{x+5}+2 $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$10^{x}-5=95$$

Complete the following. A.Use a table of \(f(x)\) and \(g(x)\) to determine whether \(f(x)=g(x) B.If possible, use properties of logarithms to show that \)f(x)=g(x)$ $$ f(x)=\ln x^{4}-\ln x^{2}, \quad g(x)=2 \ln x $$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=2-3 x, \quad g(x)=x^{3} $$

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{1}{x^{3}-1} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$2 \cdot 10^{x}=66$$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=\sqrt{x}, \quad g(x)=1-x^{2} $$

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{2}{2-x^{3}} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$10^{3 x}=100$$

Sketch a graph of \(f\) $$f(x)=\log _{2} x$$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=\frac{1}{x+1}, \quad g(x)=5 x $$

Restrict the domain of \(f(x)\) so that \(f\) is one to-one. Then find \(f^{-1}(x)\). Answers may vary. $$ f(x)=4-x^{2} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$4 \cdot 10^{2 x}+1=21$$

Sketch a graph of \(f\) $$f(x)=\log _{2} x^{2}$$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=\frac{1}{3 x}, \quad g(x)=\frac{2}{x-1} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$5\left(10^{4 x}\right)=65$$

Sketch a graph of \(f\) $$f(x)=\log _{3}|x|$$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=x+4, \quad g(x)=\sqrt{4-x^{2}} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$3\left(10^{x-2}\right)=72$$

\(\$ 3300\) at \(8 \%\) compounded quarterly for 2 years