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Intermediate Algebra

Review Exercises

Intermediate AlgebraReview Exercises
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

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)=6x11f(x)=6x11

370.

f(x)=x3+13f(x)=x3+13

371.

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

372.

f(x)=x15f(x)=x15

Evaluate and Graph Exponential Functions

Graph Exponential Functions

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

373.

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

374.

f(x)=(15)xf(x)=(15)x

375.

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

376.

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

377.

f(x)=(2.3)x3f(x)=(2.3)x3

378.

f(x)=ex+5f(x)=ex+5

379.

f(x)=exf(x)=ex

Solve Exponential Equations

In the following exercises, solve each equation.

380.

35x6=8135x6=81

381.

2x2=162x2=16

382.

9x=279x=27

383.

5x2+2x=155x2+2x=15

384.

e4x·e7=e19e4x·e7=e19

385.

ex2e15=e2xex2e15=e2x

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.

54=62554=625

391.

10−3=11,00010−3=11,000

392.

6315=6356315=635

393.

ey=16ey=16

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

394.

7=log21287=log2128

395.

5=log100,0005=log100,000

396.

4=lnx4=lnx

Evaluate Logarithmic Functions

In the following exercises, solve for x.

397.

logx125=3logx125=3

398.

log7x=−2log7x=−2

399.

log12116=xlog12116=x

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

400.

log232log232

401.

log81log81

402.

log319log319

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

403.

y=log5xy=log5x

404.

y=log14xy=log14x

405.

y=log0.8xy=log0.8x

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

406.

loga36=5loga36=5

407.

lnx=−3lnx=−3

408.

log2(5x7)=3log2(5x7)=3

409.

lne3x=24lne3x=24

410.

log(x221)=2log(x221)=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.

log4(64xy)log4(64xy)

417.

log10,000mlog10,000m

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

418.

log749ylog749y

419.

lne52lne52

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

420.

logx−9logx−9

421.

log4z7log4z7

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

422.

log3(4x7y8)log3(4x7y8)

423.

log58a2b6cd3log58a2b6cd3

424.

ln3x2y2z4ln3x2y2z4

425.

log67x26y3z53log67x26y3z53

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

426.

log256log27log256log27

427.

3log3x+7log3y3log3x+7log3y

428.

log5(x216)2log5(x+4)log5(x216)2log5(x+4)

429.

14logy2log(y3)14logy2log(y3)

Use the Change-of-Base Formula

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

430.

log597log597

431.

log316log316

Solve Exponential and Logarithmic Equations

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

432.

3log5x=log52163log5x=log5216

433.

log2x+log2(x2)=3log2x+log2(x2)=3

434.

log(x1)log(3x+5)=logxlog(x1)log(3x+5)=logx

435.

log4(x2)+log4(x+5)=log48log4(x2)+log4(x+5)=log48

436.

ln(3x2)=ln(x+4)+ln2ln(3x2)=ln(x+4)+ln2

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.

2x=1012x=101

438.

ex=23ex=23

439.

(13)x=7(13)x=7

440.

7ex+3=287ex+3=28

441.

ex4+8=23ex4+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?

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