Chapter 5

Find \(C\) and a so that \(f(x)=C a^{x}\) models the situation described. State what the variable \(x\) represents in your formula. (Answers may vary.) A fish population is initially 6000 and decreases by half each year.

Modeling Decide if the situation could be modeled by a one-to-one function. The cumulative numbers of AIDS cases from 1980 to present

Simplify the expression. $$\log _{8} 64$$

Tire Pressure The pressure in a tire with a leak is 30 pounds per square inch initially and can be modeled by \(f(x)=30(0.9)^{x}\) after \(x\) minutes. What is the tire's pressure after 9.5 minutes?

Modeling Decide if the situation could be modeled by a one-to-one function. The population of the United States from 1980 to 2000

Simplify the expression. $$\text { In } \sqrt[3] e$$

Population The population of California was about 38 million in 2007 and increasing by \(1.6 \%\) each year. Estimate the population of California in 2012 .

(Refer to Examples 5 and \(6 .\) ) Write the expression as a logarithm of a single expression. $$ \log _{3} 5-\log _{3} 10-\log _{3} \frac{1}{2} $$

Modeling Decide if the situation could be modeled by a one-to-one function. The height \(y\) of a stone thrown upward after \(x\) seconds

Simplify the expression. $$\log _{1 / 2}\left(\frac{1}{4}\right)$$

Approximate \(f(x)\) to four decimal places. $$ f(x)=4 e^{-1.2 x}, \quad x=-2.4 $$

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=2 x+1 $$

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

Simplify the expression. $$\log _{1 / 3}\left(\frac{1}{27}\right)$$

Approximate \(f(x)\) to four decimal places. $$ f(x)=-2.1 e^{-0.71 x}, \quad x=1.9 $$

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=5-\frac{1}{2} x $$

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

Simplify the expression. $$\log _{1 / 6} 36$$

Approximate \(f(x)\) to four decimal places. $$ f(x)=\frac{1}{2}\left(e^{x}-e^{-x}\right), \quad x=-0.7 $$

(Refer to Examples 5 and \(6 .\) ) Write the expression as a logarithm of a single expression. $$ \ln x^{6}-\ln x^{3} $$

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

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=-2 x+10 $$

Simplify the expression. $$\log _{1 / 4} 64$$

Approximate \(f(x)\) to four decimal places. $$ f(x)=4\left(e^{-0.3 x}-e^{-0.6 x}\right), \quad x=1.6 $$

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=-(x-1)^{2} $$

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

Simplify the expression. $$\log _{a} \frac{1}{a}$$

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

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=\frac{1}{2} x^{2}+3 x-1 $$

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

Simplify the expression. $$\log _{a}\left(a^{2} \cdot a^{3}\right)$$

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

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=2 x^{2}-x-9 $$

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

Sketch a graph of \(y=f(x)\) $$ f(x)=3^{-x} $$

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

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=2 x^{3}-5 $$

Simplify the expression. $$\text { ln } \sqrt{e}$$

Sketch a graph of \(y=f(x)\) $$ f(x)=3\left(2^{-x}\right) $$

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

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

Simplify the expression. $$\log _{2} \frac{1}{16}$$

Sketch a graph of \(y=f(x)\) $$ f(x)=2\left(\frac{1}{3}\right)^{x} $$

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=|3 x-1|+4 $$

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=x^{2}-1, x \geq 0 $$

Simplify the expression. $$\log _{8} 8^{k}$$

Sketch a graph of \(y=f(x)\) $$ f(x)=2\left(3^{x}\right) $$

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=2|1-x|-7 $$

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

Sketch a graph of \(y=f(x)\) $$ f(x)=\left(\frac{1}{2}\right)^{x} $$