### Review Exercises

## Finding Composite and Inverse Functions

**Find and Evaluate Composite Functions**

In the following exercises, for each pair of functions, find ⓐ (*f* ∘ *g*)(*x*), ⓑ (*g* ∘ *f*)(*x*), and ⓒ (*f* · *g*)(*x*).

$f\left(x\right)=7x-2$ and

$g\left(x\right)=5x+1$

In the following exercises, evaluate the composition.

For functions

$f\left(x\right)=3{x}^{2}+2$ and

$g\left(x\right)=4x-3,$ find

ⓐ
$\left(f\circ g\right)\left(\mathrm{-3}\right)$

ⓑ
$\left(g\circ f\right)\left(\mathrm{-2}\right)$

ⓒ
$\left(f\circ f\right)\left(\mathrm{-1}\right)$

For functions

$f\left(x\right)=2{x}^{3}+5$ and

$g\left(x\right)=3{x}^{2}-7,$ find

ⓐ
$\left(f\circ g\right)\left(\mathrm{-1}\right)$

ⓑ
$\left(g\circ f\right)\left(\mathrm{-2}\right)$

ⓒ
$\left(g\circ g\right)\left(1\right)$

**Determine Whether a Function is One-to-One**

In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

$\{\left(\mathrm{-3},\mathrm{-5}\right),\left(\mathrm{-2},\mathrm{-4}\right),\left(\mathrm{-1},\mathrm{-3}\right),\left(0,\mathrm{-2}\right),$

$\left(\mathrm{-1},\mathrm{-1}\right),\left(\mathrm{-2},0\right),\left(\mathrm{-3},1\right)\}$

$\{\left(\mathrm{-3},0\right),\left(\mathrm{-2},\mathrm{-2}\right),\left(\mathrm{-1},0\right),\left(0,1\right),$

$\left(1,2\right),\left(2,1\right),\left(3,\mathrm{-1}\right)\}$

$\{\left(\mathrm{-3},3\right),\left(\mathrm{-2},1\right),\left(\mathrm{-1},\mathrm{-1}\right),\left(0,\mathrm{-3}\right),$

$\left(1,\mathrm{-5}\right),\left(2,\mathrm{-4}\right),\left(3,\mathrm{-2}\right)\}$

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

ⓐ

ⓑ

**Find the Inverse of a Function**

In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.

$\left\{\left(\mathrm{-3},10\right),\left(\mathrm{-2},5\right),\left(\mathrm{-1},2\right),\left(0,1\right)\right\}$

In the following exercise, graph the inverse of the one-to-one function shown.

In the following exercises, verify that the functions are inverse functions.

$f\left(x\right)=2x+9$ and

$g\left(x\right)=\frac{x+9}{2}$

In the following exercises, find the inverse of each function.

$f\left(x\right)={x}^{3}+13$

$f\left(x\right)=\sqrt[5]{x-1}$

## Evaluate and Graph Exponential Functions

**Graph Exponential Functions**

In the following exercises, graph each of the following functions.

$f\left(x\right)={\left(\frac{1}{5}\right)}^{x}$

$g\left(x\right)={3}^{x+2}$

$f\left(x\right)={e}^{x}+5$

**Solve Exponential Equations**

In the following exercises, solve each equation.

${3}^{5x-6}=81$

${9}^{x}=27$

${e}^{4x}\xb7{e}^{7}={e}^{19}$

**Use Exponential Models in Applications**

In the following exercises, solve.

Felix invested $\text{\$}\mathrm{12,000}$ in a savings account. If the interest rate is $4\text{\%}$ how much will be in the account in 12 years by each method of compounding?

ⓐ compound quarterly

ⓑ compound monthly

ⓒ compound continuously.

Sayed deposits $\text{\$}\mathrm{20,000}$ in an investment account. What will be the value of his investment in 30 years if the investment is earning $7\text{\%}$ per year and is compounded continuously?

A researcher at the Center for Disease Control and Prevention is studying the growth of a bacteria. She starts her experiment with 150 of the bacteria that grows at a rate of $15\text{\%}$ per hour. She will check on the bacteria every 24 hours. How many bacteria will he find in 24 hours?

In the last five years the population of the United States has grown at a rate of $0.7\text{\%}$ per year to about 318,900,000. If this rate continues, what will be the population in 5 more years?

## Evaluate and Graph Logarithmic Functions

**Convert Between Exponential and Logarithmic Form**

In the following exercises, convert from exponential to logarithmic form.

${5}^{4}=625$

${63}^{\frac{1}{5}}=\sqrt[5]{63}$

In the following exercises, convert each logarithmic equation to exponential form.

$7={\text{log}}_{2}128$

$4=\text{ln}\phantom{\rule{0.2em}{0ex}}x$

**Evaluate Logarithmic Functions**

In the following exercises, solve for *x*.

${\text{log}}_{7}x=\mathrm{-2}$

In the following exercises, find the exact value of each logarithm without using a calculator.

${\text{log}}_{2}32$

${\text{log}}_{3}\frac{1}{9}$

**Graph Logarithmic Functions**

In the following exercises, graph each logarithmic function.

$y={\text{log}}_{\frac{1}{4}}x$

**Solve Logarithmic Equations**

In the following exercises, solve each logarithmic equation.

${\text{log}}_{a}36=5$

${\text{log}}_{2}\left(5x-7\right)=3$

$\text{log}({x}^{2}-21)=2$

**Use Logarithmic Models in Applications**

What is the decibel level of a train whistle with intensity ${10}^{\mathrm{-3}}$ watts per square inch?

## Use the Properties of Logarithms

**Use the Properties of Logarithms**

In the following exercises, use the properties of logarithms to evaluate.

ⓐ ${\text{log}}_{7}1$ ⓑ ${\text{log}}_{12}12$

ⓐ ${10}^{\text{log}\sqrt{5}}$ ⓑ $\text{log}{10}^{\mathrm{-3}}$

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

${\text{log}}_{4}\left(64xy\right)$

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

${\text{log}}_{7}\frac{49}{y}$

In the following exercises, use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.

$\text{log}{x}^{\mathrm{-9}}$

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

${\text{log}}_{3}\left(\sqrt{4}{x}^{7}{y}^{8}\right)$

$\text{ln}\frac{\sqrt{3{x}^{2}-{y}^{2}}}{{z}^{4}}$

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

${\text{log}}_{2}56-{\text{log}}_{2}7$

${\text{log}}_{5}\left({x}^{2}-16\right)-2{\text{log}}_{5}\left(x+4\right)$

**Use the Change-of-Base Formula**

In the following exercises, rounding to three decimal places, approximate each logarithm.

${\text{log}}_{5}97$

## Solve Exponential and Logarithmic Equations

**Solve Logarithmic Equations Using the Properties of Logarithms**

In the following exercises, solve for *x*.

$3{\text{log}}_{5}x={\text{log}}_{5}216$

$\text{log}\left(x-1\right)-\text{log}\left(3x+5\right)=\text{\u2212}\text{log}\phantom{\rule{0.2em}{0ex}}x$

$\text{ln}\left(3x-2\right)=\text{ln}\left(x+4\right)+\text{ln}2$

**Solve Exponential Equations Using Logarithms**

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

${e}^{x}=23$

$7{e}^{x+3}=28$

**Use Exponential Models in Applications**

Jerome invests $\text{\$}\mathrm{18,000}$ at age 17. He hopes the investments will be worth $\text{\$}\mathrm{30,000}$ when he turns 26. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?

Elise invests $\text{\$}4500$ in an account that compounds interest monthly and earns $6\text{\%}.$ How long will it take for her money to double?

Researchers recorded that a certain bacteria population grew from 100 to 300 in 8 hours. At this rate of growth, how many bacteria will there be in 24 hours?

Mouse populations can double in 8 months $\left(A=2{A}_{0}\right).$ How long will it take for a mouse population to triple?

The half-life of radioactive iodine is 60 days. How much of a 50 mg sample will be left in 40 days?