20.7: Review Exercises

Chapter Review Exercises

Finding Composite and Inverse Functions

Exercise $\PageIndex{1}$ Find and Evaluate Composite Functions

In the following exercises, for each pair of functions, find

1. $(f \circ g)(x)$
2. $(g \circ f)(x)$
3. $(f \cdot g)(x)$

1. $f(x)=7 x-2$ and $g(x)=5 x+1$

2. $f(x)=4 x$ and $g(x)=x^{2}+3 x$

Answer

2.

1. $4 x^{2}+12 x$
2. $16 x^{2}+12 x$
3. $4 x^{3}+12 x^{2}$

Exercise $\PageIndex{2}$ Find and Evaluate Composite Functions

In the following exercises, evaluate the composition.

1. For functions $f(x)=3 x^{2}+2$ and $g(x)=4 x-3$, find
   1. $(f \circ g)(-3)$
   2. $(g \circ f)(-2)$
   3. $(f \circ f)(-1)$
2. For functions $f(x)=2 x^{3}+5$ and $g(x)=3 x^{2}-7$, find
   1. $(f \circ g)(-1)$
   2. $(g \circ f)(-2)$
   3. $(g \circ g)(1)$

Answer

2.

1. $-123$
2. $356$
3. $41$

Exercise $\PageIndex{3}$ 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.

1. $\begin{array}{l}{\{(-3,-5),(-2,-4),(-1,-3),(0,-2)} , {(-1,-1),(-2,0),(-3,1) \}}\end{array}$
2. $\begin{array}{l}{\{(-3,0),(-2,-2),(-1,0),(0,1)} , {(1,2),(2,1),(3,-1) \}}\end{array}$
3. $\begin{array}{l}{\{(-3,3),(-2,1),(-1,-1),(0,-3)} , {(1,-5),(2,-4),(3,-2) \}}\end{array}$

Answer

2. Function; not one-to-one

Exercise $\PageIndex{4}$ Determine Whether a Function is One-to-One

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

1. 1. 
      
      Figure 10.E.1
   2. 
      
      Figure 10.E.2
2. 1. 
      
      Figure 10.E.3
   2. 
      
      Figure 10.E.4

Answer

1.

1. Function; not one-to-one
2. Not a function

Exercise $\PageIndex{5}$ 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.

1. $\{(-3,10),(-2,5),(-1,2),(0,1)\}$

Answer

1. Inverse function: $\{(10,-3),(5,-2),(2,-1),(1,0)\}$. Domain: $\{1,2,5,10\}$. Range: $\{-3,-2,-1,0\}$.

Exercise $\PageIndex{6}$ Find the Inverse of a Function

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

Figure 10.E.5

Answer

Solve on your own

Exercise $\PageIndex{7}$ Find the Inverse of a Function

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

1. $\begin{array}{l}{f(x)=3 x+7 \text { and }} {g(x)=\frac{x-7}{3}}\end{array}$
2. $\begin{array}{l}{f(x)=2 x+9 \text { and }} {g(x)=\frac{x+9}{2}}\end{array}$

Answer

1. $g(f(x))=x,$ and $f(g(x))=x,$ so they are inverses.

Exercise $\PageIndex{8}$ Find the Inverse of a Function

1. $f(x)=6 x-11$
2. $f(x)=x^{3}+13$
3. $f(x)=\frac{1}{x+5}$
4. $f(x)=\sqrt[5]{x-1}$

Answer

1. $f^{-1}(x)=\frac{x+11}{6}$

3. $f^{-1}(x)=\frac{1}{x}-5$

Evaluate and Graph Exponential Functions

Exercise $\PageIndex{9}$ Graph Exponential Functions

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

1. $f(x)=4^{x}$
2. $f(x)=\left(\frac{1}{5}\right)^{x}$
3. $g(x)=(0.75)^{x}$
4. $g(x)=3^{x+2}$
5. $f(x)=(2.3)^{x}-3$
6. $f(x)=e^{x}+5$
7. $f(x)=-e^{x}$

Answer

1.

Figure 10.E.6

3.

Figure 10.E.7

5.

Figure 10.E.8

7.

Figure 10.E.9

Exercise $\PageIndex{10}$ Solve Exponential Equations

In the following exercises, solve each equation.

1. $3^{5 x-6}=81$
2. $2^{x^{2}}=16$
3. $9^{x}=27$
4. $5^{x^{2}+2 x}=\frac{1}{5}$
5. $e^{4 x} \cdot e^{7}=e^{19}$
6. $\frac{e^{x^{2}}}{e^{15}}=e^{2 x}$

Answer

2. $x=-2, x=2$

4. $x=-1$

6. $x=-3, x=5$

Exercise $\PageIndex{11}$ Use Exponential Models in Applications

In the following exercises, solve.

1. Felix invested $$12,000$ in a savings account. If the interest rate is $4$% how much will be in the account in $12$ years by each method of compounding?
   1. compound quarterly
   2. compound monthly
   3. compound continuously
2. Sayed deposits $$20,000$ in an investment account. What will be the value of his investment in $30$ years if the investment is earning $7$% per year and is compounded continuously?
3. 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 $15$% per hour. She will check on the bacteria every $24$ hours. How many bacteria will he find in $24$ hours?
4. In the last five years the population of the United States has grown at a rate of $0.7$% per year to about $318,900,000$. If this rate continues, what will be the population in $5$ more years?

Answer

2. $\$ 163,323.40$

4. $330,259,000$

Evaluate and Graph Logarithmic Functions

Exercise $\PageIndex{12}$ Convert Between Exponential and Logarithmic Form

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

1. $5^{4}=625$
2. $10^{-3}=\frac{1}{1,000}$
3. $63^{\frac{1}{5}}=\sqrt[5]{63}$
4. $e^{y}=16$

Answer

2. $\log \frac{1}{1,000}=-3$

4. $\ln 16=y$

Exercise $\PageIndex{13}$ Convert Between Exponential and Logarithmic Form

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

1. $7=\log _{2} 128$
2. $5=\log 100,000$
3. $4=\ln x$

Answer

2. $100000=10^{5}$

Exercise $\PageIndex{14}$ Evaluate Logarithmic Functions

In the following exercises, solve for $x$.

1. $\log _{x} 125=3$
2. $\log _{7} x=-2$
3. $\log _{\frac{1}{2}} \frac{1}{16}=x$

Answer

1. $x=5$

3. $x=4$

Exercise $\PageIndex{15}$ Evaluate Logarithmic Functions

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

1. $\log _{2} 32$
2. $\log _{8} 1$
3. $\log _{3} \frac{1}{9}$

Answer

2. $0$

Exercise $\PageIndex{16}$ Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

1. $y=\log _{5} x$
2. $y=\log _{\frac{1}{4}} x$
3. $y=\log _{0.8} x$

Answer

1.

Figure 10.E.10

3.

Figure 10.E.11

Exercise $\PageIndex{17}$ Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

1. $\log _{a} 36=5$
2. $\ln x=-3$
3. $\log _{2}(5 x-7)=3$
4. $\ln e^{3 x}=24$
5. $\log \left(x^{2}-21\right)=2$

Answer

2. $x=e^{-3}$

4. $x=8$

Exercise $\PageIndex{18}$ Use Logarithmic Models in Applications

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

Answer

$90$ dB

Use the Properties of Logarithms

Exercise $\PageIndex{19}$ Use the Properties of Logarithms

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

1. 1. $\log _{7} 1$
   2. $\log _{12} 12$
2. 1. $5^{\log _{5} 13}$
   2. $\log _{3} 3^{-9}$
3. 1. $10^{\log \sqrt{5}}$
   2. $\log 10^{-3}$
4. 1. $e^{\ln 8}$
   2. $\ln e^{5}$

Answer

2.

1. $13$
2. $-9$

4.

1. $8$
2. $5$

Exercise $\PageIndex{20}$ Use the Properties of Logarithms

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

1. $\log _{4}(64 x y)$
2. $\log 10,000 m$

Answer

2. $4+\log m$

Exercise $\PageIndex{21}$ Use the Properties of Logarithms

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

1. $\log _{7} \frac{49}{y}$
2. $\ln \frac{e^{5}}{2}$

Answer

2. $5-\ln 2$

Exercise $\PageIndex{22}$ Use the Properties of Logarithms

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

1. $\log x^{-9}$
2. $\log _{4} \sqrt[7]{z}$

Answer

2. $\frac{1}{7} \log _{4} z$

Exercise $\PageIndex{23}$ Use the Properties of Logarithms

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

1. $\log _{3}\left(\sqrt{4} x^{7} y^{8}\right)$
2. $\log _{5} \frac{8 a^{2} b^{6} c}{d^{3}}$
3. $\ln \frac{\sqrt{3 x^{2}-y^{2}}}{z^{4}}$
4. $\log _{6} \sqrt[3]{\frac{7 x^{2}}{6 y^{3} z^{5}}}$

Answer

2. $\begin{array}{l}{\log _{5} 8+2 \log _{5} a+6 \log _{5} b} {+\log _{5} c-3 \log _{5} d}\end{array}$

4. $\begin{array}{l}{\frac{1}{3}\left(\log _{6} 7+2 \log _{6} x-1-3 \log _{6} y\right.} {-5 \log _{6} z )}\end{array}$

Exercise $\PageIndex{24}$ Use the Properties of Logarithms

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

1. $\log _{2} 56-\log _{2} 7$
2. $3 \log _{3} x+7 \log _{3} y$
3. $\log _{5}\left(x^{2}-16\right)-2 \log _{5}(x+4)$
4. $\frac{1}{4} \log y-2 \log (y-3)$

Answer

2. $\log _{3} x^{3} y^{7}$

4. $\log \frac{\sqrt[4]{y}}{(y-3)^{2}}$

Exercise $\PageIndex{25}$ Use the Change-of-Base Formula

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

1. $\log _{5} 97$
2. $\log _{\sqrt{3}} 16$

Answer

2. $5.047$

Solve Exponential and Logarithmic Equations

Exercise $\PageIndex{26}$ Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for $x$.

1. $3 \log _{5} x=\log _{5} 216$
2. $\log _{2} x+\log _{2}(x-2)=3$
3. $\log (x-1)-\log (3 x+5)=-\log x$
4. $\log _{4}(x-2)+\log _{4}(x+5)=\log _{4} 8$
5. $\ln (3 x-2)=\ln (x+4)+\ln 2$

Answer

2. $x=4$

4. $x=3$

Exercise $\PageIndex{27}$ 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.

1. $2^{x}=101$
2. $e^{x}=23$
3. $\left(\frac{1}{3}\right)^{x}=7$
4. $7 e^{x+3}=28$
5. $e^{x-4}+8=23$

Answer

1. $x=\frac{\log 101}{\log 2} \approx 6.658$

3. $x=\frac{\log 7}{\log \frac{1}{3}} \approx-1.771$

5. $x=\ln 15+4 \approx 6.708$

Exercise $\PageIndex{28}$ Use Exponential Models in Applications

1. Jerome invests $$18,000$ at age $17$. He hopes the investments will be worth $$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?
2. Elise invests $$4500$ in an account that compounds interest monthly and earns $6$%.How long will it take for her money to double?
3. 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?
4. Mouse populations can double in $8$ months $\left(A=2 A_{0}\right)$. How long will it take for a mouse population to triple?
5. The half-life of radioactive iodine is $60$ days. How much of a $50$ mg sample will be left in $40$ days?

Answer

2. $11.6$ years

4. $12.7$ months

Practice Test

Exercise $\PageIndex{29}$

1. For the functions, $f(x)=6x+1$ and $g(x)=8x−3$, find
   1. $(f \circ g)(x)$
   2. $(g \circ f)(x)$
   3. $(f \cdot g)(x)$
2. Determine if the following set of ordered pairs represents a function and if so, is the function one-to-one. $\{(-2,2),(-1,-3),(0,1),(1,-2),(2,-3)\}$
3. Determine whether each graph is the graph of a function and if so, is it one-to-one.
   1. 
      
      Figure 10.E.12
   2. 
      
      Figure 10.E.13
4. Graph, on the same coordinate system, the inverse of the one-to-one function shown.

Figure 10.E.14

5. Find the inverse of the function $f(x)=x^{5}−9$.

6. Graph the function $g(x)=2^{x-3}$.

7. Solve the equation $2^{2 x-4}=64$.

8. Solve the equation $\frac{e^{x^{2}}}{e^{4}}=e^{3 x}$.

9. Megan invested $$21,000$ in a savings account. If the interest rate is $5$%, how much will be in the account in $8$ years by each method of compounding?

1. compound quarterly
2. compound monthly
3. compound continuously

10. Convert the equation from exponential to logarithmic form: $10^{-2}=\frac{1}{100}$.

11. Convert the equation from logarithmic equation to exponential form: $3=\log _{7} 343$.

12. Solve for $x$: $\log _{5} x=-3$

13. Evaluate log $_{11} 1$.

14. Evaluate $\log _{4} \frac{1}{64}$.

15. Graph the function $y=\log _{3} x$.

16. Solve for $x$: $\log \left(x^{2}-39\right)=1$

17. What is the decibel level of a small fan with intensity $10^{−8}$ watts per square inch?

18. Evaluate each.

1. $6^{\log _{6} 17}$
2. $\log _{9} 9^{-3}$

Answer

1.

1. $48 x-17$
2. $48 x+5$
3. $48 x^{2}-10 x-3$

3.

1. Not a function
2. One-to-one function

5. $f^{-1}(x)=\sqrt[5]{x+9}$

7. $x=5$

9.

1. $$31,250.74$
2. $$31,302.29$
3. $$31,328.32$

11. $343=7^{3}$

13. $0$

15.

Figure 10.E.15

17. $40$ dB

Exercise $\PageIndex{30}$

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

1. $\log _{5} 25 a b$
2. $\ln \frac{e^{12}}{8}$
3. $\log _{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}}$

Answer

1. $2+\log _{5} a+\log _{5} b$

3. $\begin{array}{l}{\frac{1}{4}\left(\log _{2} 5+3 \log _{2} x-4-2 \log _{2} y\right.} {-7 \log _{2} z )}\end{array}$

Exercise $\PageIndex{31}$

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

1. $5 \log _{4} x+3 \log _{4} y$
2. $\frac{1}{6} \log x-3 \log (x+5)$
3. Rounding to three decimal places, approximate $\log _{4} 73$.
4. Solve for $x$: $\log _{7}(x+2)+\log _{7}(x-3)=\log _{7} 24$

Answer

2. $\log \frac{\sqrt[6]{x}}{(x+5)^{3}}$

4. $x=6$

Exercise $\PageIndex{32}$

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

1. $\left(\frac{1}{5}\right)^{x}=9$
2. $5 e^{x-4}=40$
3. Jacob invests $$14,000$ in an account that compounds interest quarterly and earns $4$%. How long will it take for his money to double?
4. Researchers recorded that a certain bacteria population grew from $500$ to $700$ in $5$ hours. At this rate of growth, how many bacteria will there be in $20$ hours?
5. A certain beetle population can double in $3$ months $\left(A=2 A_{0}\right)$. How long will it take for that beetle population to triple?

Answer

2. $x=\ln 8+4 \approx 6.079$

4. $1,921$ bacteria