Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Intermediate Algebra 2e

Review Exercises

Intermediate Algebra 2eReview Exercises

Review Exercises

Finding Composite and Inverse Functions

Find and Evaluate Composite Functions

In the following exercises, for each pair of functions, find (fg)(x), (gf)(x), and (f · g)(x).

356.

f(x)=7x2f(x)=7x2 and
g(x)=5x+1g(x)=5x+1

357.

f(x)=4xf(x)=4x and
g(x)=x2+3xg(x)=x2+3x

In the following exercises, evaluate the composition.

358.

For functions
f(x)=3x2+2f(x)=3x2+2 and
g(x)=4x3,g(x)=4x3, find
(fg)(−3)(fg)(−3)
(gf)(−2)(gf)(−2)
(ff)(−1)(ff)(−1)

359.

For functions
f(x)=2x3+5f(x)=2x3+5 and
g(x)=3x27,g(x)=3x27, find
(fg)(−1)(fg)(−1)
(gf)(−2)(gf)(−2)
(gg)(1)(gg)(1)

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.

360.

{ ( −3 , −5 ) , ( −2 , −4 ) , ( −1 , −3 ) , ( 0 , −2 ) , { ( −3 , −5 ) , ( −2 , −4 ) , ( −1 , −3 ) , ( 0 , −2 ) ,
( −1 , −1 ) , ( −2 , 0 ) , ( −3 , 1 ) } ( −1 , −1 ) , ( −2 , 0 ) , ( −3 , 1 ) }

361.

{ ( −3 , 0 ) , ( −2 , −2 ) , ( −1 , 0 ) , ( 0 , 1 ) , { ( −3 , 0 ) , ( −2 , −2 ) , ( −1 , 0 ) , ( 0 , 1 ) ,
( 1 , 2 ) , ( 2 , 1 ) , ( 3 , −1 ) } ( 1 , 2 ) , ( 2 , 1 ) , ( 3 , −1 ) }

362.

{ ( −3 , 3 ) , ( −2 , 1 ) , ( −1 , −1 ) , ( 0 , −3 ) , { ( −3 , 3 ) , ( −2 , 1 ) , ( −1 , −1 ) , ( 0 , −3 ) ,
( 1 , −5 ) , ( 2 , −4 ) , ( 3 , −2 ) } ( 1 , −5 ) , ( 2 , −4 ) , ( 3 , −2 ) }

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

363.


This figure shows a line from (negative 6, negative 2) up to (negative 1, 3) and then down from there to (6, negative 4).



This figure shows a line from (6, 5) down to (0, negative 1) and then down from there to (5, negative 6).
364.


This figure shows a curved line from (negative 6, negative 2) up to the origin and then continuing up from there to (6, 2).



This figure shows a circle of radius 2 with center at the origin.

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.

365.

{ ( −3 , 10 ) , ( −2 , 5 ) , ( −1 , 2 ) , ( 0 , 1 ) } { ( −3 , 10 ) , ( −2 , 5 ) , ( −1 , 2 ) , ( 0 , 1 ) }

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

366.
This figure shows a line segment from (negative 4, negative 2) up to (negative 2, 1) then up to (2, 2) and then up to (3, 4).

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

367.

f(x)=3x+7f(x)=3x+7 and
g(x)=x73g(x)=x73

368.

f(x)=2x+9f(x)=2x+9 and
g(x)=x+92g(x)=x+92

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

369.

f ( x ) = 6 x 11 f ( x ) = 6 x 11

370.

f ( x ) = x 3 + 13 f ( x ) = x 3 + 13

371.

f ( x ) = 1 x + 5 f ( x ) = 1 x + 5

372.

f ( x ) = x 1 5 f ( x ) = x 1 5

Evaluate and Graph Exponential Functions

Graph Exponential Functions

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

373.

f ( x ) = 4 x f ( x ) = 4 x

374.

f ( x ) = ( 1 5 ) x f ( x ) = ( 1 5 ) x

375.

g ( x ) = ( 0.75 ) x g ( x ) = ( 0.75 ) x

376.

g ( x ) = 3 x + 2 g ( x ) = 3 x + 2

377.

f ( x ) = ( 2.3 ) x 3 f ( x ) = ( 2.3 ) x 3

378.

f ( x ) = e x + 5 f ( x ) = e x + 5

379.

f ( x ) = e x f ( x ) = e x

Solve Exponential Equations

In the following exercises, solve each equation.

380.

3 5 x 6 = 81 3 5 x 6 = 81

381.

2 x 2 = 16 2 x 2 = 16

382.

9 x = 27 9 x = 27

383.

5 x 2 + 2 x = 1 5 5 x 2 + 2 x = 1 5

384.

e 4 x · e 7 = e 19 e 4 x · e 7 = e 19

385.

e x 2 e 15 = e 2 x e x 2 e 15 = e 2 x

Use Exponential Models in Applications

In the following exercises, solve.

386.

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

compound quarterly
compound monthly
compound continuously.

387.

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

388.

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%15% per hour. She will check on the bacteria every 24 hours. How many bacteria will he find in 24 hours?

389.

In the last five years the population of the United States has grown at a rate of 0.7%0.7% 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.

390.

5 4 = 625 5 4 = 625

391.

10 −3 = 1 1,000 10 −3 = 1 1,000

392.

63 1 5 = 63 5 63 1 5 = 63 5

393.

e y = 16 e y = 16

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

394.

7 = log 2 128 7 = log 2 128

395.

5 = log 100,000 5 = log 100,000

396.

4 = ln x 4 = ln x

Evaluate Logarithmic Functions

In the following exercises, solve for x.

397.

log x 125 = 3 log x 125 = 3

398.

log 7 x = −2 log 7 x = −2

399.

log 1 2 1 16 = x log 1 2 1 16 = x

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

400.

log 2 32 log 2 32

401.

log 8 1 log 8 1

402.

log 3 1 9 log 3 1 9

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

403.

y = log 5 x y = log 5 x

404.

y = log 1 4 x y = log 1 4 x

405.

y = log 0.8 x y = log 0.8 x

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

406.

log a 36 = 5 log a 36 = 5

407.

ln x = −3 ln x = −3

408.

log 2 ( 5 x 7 ) = 3 log 2 ( 5 x 7 ) = 3

409.

ln e 3 x = 24 ln e 3 x = 24

410.

log ( x 2 21 ) = 2 log ( x 2 21 ) = 2

Use Logarithmic Models in Applications

411.

What is the decibel level of a train whistle with intensity 10−310−3 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.

412.

log71log71 log1212log1212

413.

5log5135log513 log33−9log33−9

414.

10log510log5 log10−3log10−3

415.

eln8eln8 lne5lne5

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

416.

log 4 ( 64 x y ) log 4 ( 64 x y )

417.

log 10,000 m log 10,000 m

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

418.

log 7 49 y log 7 49 y

419.

ln e 5 2 ln e 5 2

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

420.

log x −9 log x −9

421.

log 4 z 7 log 4 z 7

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

422.

log 3 ( 4 x 7 y 8 ) log 3 ( 4 x 7 y 8 )

423.

log 5 8 a 2 b 6 c d 3 log 5 8 a 2 b 6 c d 3

424.

ln 3 x 2 y 2 z 4 ln 3 x 2 y 2 z 4

425.

log 6 7 x 2 6 y 3 z 5 3 log 6 7 x 2 6 y 3 z 5 3

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

426.

log 2 56 log 2 7 log 2 56 log 2 7

427.

3 log 3 x + 7 log 3 y 3 log 3 x + 7 log 3 y

428.

log 5 ( x 2 16 ) 2 log 5 ( x + 4 ) log 5 ( x 2 16 ) 2 log 5 ( x + 4 )

429.

1 4 log y 2 log ( y 3 ) 1 4 log y 2 log ( y 3 )

Use the Change-of-Base Formula

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

430.

log 5 97 log 5 97

431.

log 3 16 log 3 16

Solve Exponential and Logarithmic Equations

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

432.

3 log 5 x = log 5 216 3 log 5 x = log 5 216

433.

log 2 x + log 2 ( x 2 ) = 3 log 2 x + log 2 ( x 2 ) = 3

434.

log ( x 1 ) log ( 3 x + 5 ) = log x log ( x 1 ) log ( 3 x + 5 ) = log x

435.

log 4 ( x 2 ) + log 4 ( x + 5 ) = log 4 8 log 4 ( x 2 ) + log 4 ( x + 5 ) = log 4 8

436.

ln ( 3 x 2 ) = ln ( x + 4 ) + ln 2 ln ( 3 x 2 ) = ln ( x + 4 ) + 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.

437.

2 x = 101 2 x = 101

438.

e x = 23 e x = 23

439.

( 1 3 ) x = 7 ( 1 3 ) x = 7

440.

7 e x + 3 = 28 7 e x + 3 = 28

441.

e x 4 + 8 = 23 e x 4 + 8 = 23

Use Exponential Models in Applications

442.

Jerome invests $18,000$18,000 at age 17. He hopes the investments will be worth $30,000$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?

443.

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

444.

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?

445.

Mouse populations can double in 8 months (A=2A0).(A=2A0). How long will it take for a mouse population to triple?

446.

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

Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
Citation information

© Jan 23, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.