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Intermediate Algebra 2e

Key Concepts

Intermediate Algebra 2eKey Concepts
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Table of contents
  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. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    7. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. 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 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    7. 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

Key Concepts

10.1 Finding Composite and Inverse Functions

  • Composition of Functions: The composition of functions ff and g,g, is written f∘gf∘g and is defined by
    (f∘g)(x)=f(g(x))(f∘g)(x)=f(g(x))

     We read f(g(x))f(g(x)) as ff of gg of x.x.
  • Horizontal Line Test: If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.
  • Inverse of a Function Defined by Ordered Pairs: If f(x)f(x) is a one-to-one function whose ordered pairs are of the form (x,y),(x,y), then its inverse function f−1(x)f−1(x) is the set of ordered pairs (y,x).(y,x).
  • Inverse Functions: For every xx in the domain of one-to-one function ff and f−1,f−1,
    f−1(f(x))=xf(f−1(x))=xf−1(f(x))=xf(f−1(x))=x
  • How to Find the Inverse of a One-to-One Function:
    1. Step 1. Substitute y for f(x).f(x).
    2. Step 2. Interchange the variables x and y.
    3. Step 3. Solve for y.
    4. Step 4. Substitute f−1(x)f−1(x) for y.y.
    5. Step 5. Verify that the functions are inverses.

10.2 Evaluate and Graph Exponential Functions

  • Properties of the Graph of f(x)=ax:f(x)=ax:
    when a>1a>1 when 0<a<10<a<1
    Domain (−∞,∞)(−∞,∞) Domain (−∞,∞)(−∞,∞)
    Range (0,∞)(0,∞) Range (0,∞)(0,∞)
    xx-intercept none xx-intercept none
    yy-intercept (0,1)(0,1) yy-intercept (0,1)(0,1)
    Contains (1,a),(−1,1a)(1,a),(−1,1a) Contains (1,a),(−1,1a)(1,a),(−1,1a)
    Asymptote xx-axis, the line y=0y=0 Asymptote xx-axis, the line y=0y=0
    Basic shape increasing Basic shape decreasing

    This figure has two parts. On the left, we have a curve that passes through (negative 1, 1 over a) through (0, 1) to (1, a). On the right, where a is noted to be less than 1, we have a curve that passes through (negative 1, 1 over a) through (0, 1) to (1, a).
  • One-to-One Property of Exponential Functions:
     For a>0a>0 and a≠1,a≠1,
    Ifax=ay,thenx=y.Ifax=ay,thenx=y.
  • How to Solve an Exponential Equation
    1. Step 1. Write both sides of the equation with the same base, if possible.
    2. Step 2. Write a new equation by setting the exponents equal.
    3. Step 3. Solve the equation.
    4. Step 4. Check the solution.
  • Compound Interest: For a principal, P,P, invested at an interest rate, r,r, for tt years, the new balance, A,A, is
     A=P(1+rn)ntwhen compoundedntimes a year. A=Pertwhen compounded continuously. A=P(1+rn)ntwhen compoundedntimes a year. A=Pertwhen compounded continuously.
  • Exponential Growth and Decay: For an original amount, A0A0 that grows or decays at a rate, r,r, for a certain time t,t, the final amount,A,A, is A=A0ert.A=A0ert.

10.3 Evaluate and Graph Logarithmic Functions

  • Properties of the Graph of y=logax:y=logax:
    when a>1a>1 when 0<a<10<a<1
    Domain (0,∞)(0,∞) Domain (0,∞)(0,∞)
    Range (−∞,∞)(−∞,∞) Range (−∞,∞)(−∞,∞)
    x-intercept (1,0)(1,0) x-intercept (1,0)(1,0)
    y-intercept none y-intercept none
    Contains (a,1),(a,1),(1a,−1)(1a,−1) Contains (a,1),(a,1),(1a,−1)(1a,−1)
    Asymptote y-axis Asymptote y-axis
    Basic shape increasing Basic shape decreasing

    This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).
  • Decibel Level of Sound: The loudness level, DD, measured in decibels, of a sound of intensity, II, measured in watts per square inch is D=10log(I10−12).D=10log(I10−12).
  • Earthquake Intensity: The magnitude RR of an earthquake is measured by R=logI,R=logI, where II is the intensity of its shock wave.

10.4 Use the Properties of Logarithms

  • Properties of Logarithms
    loga1=0logaa=1loga1=0logaa=1
  • Inverse Properties of Logarithms
    • For a>0,a>0,x>0x>0 and a≠1a≠1
      alogax=xlogaax=xalogax=xlogaax=x
  • Product Property of Logarithms
    • If M>0,N>0,a>0M>0,N>0,a>0 and a≠1,a≠1, then,
      logaM·N=logaM+logaNlogaM·N=logaM+logaN

       The logarithm of a product is the sum of the logarithms.
  • Quotient Property of Logarithms
    • If M>0,N>0,a>0M>0,N>0,a>0 and a≠1,a≠1, then,
      logaMN=logaM−logaNlogaMN=logaM−logaN

       The logarithm of a quotient is the difference of the logarithms.
  • Power Property of Logarithms
    • If M>0,a>0,a≠1M>0,a>0,a≠1 and pp is any real number then,
      logaMp=plogaMlogaMp=plogaM

       The log of a number raised to a power is the product of the power times the log of the number.
  • Properties of Logarithms Summary
     If M>0,a>0,a≠1M>0,a>0,a≠1 and pp is any real number then,
    Property Base aa Base ee
    loga1=0loga1=0 ln1=0ln1=0
    logaa=1logaa=1 lne=1lne=1
    Inverse Properties alogax=x logaax=xalogax=x logaax=x elnx=x lnex=xelnx=x lnex=x
    Product Property of Logarithms loga(M·N)=logaM+logaNloga(M·N)=logaM+logaN ln(M·N)=lnM+lnNln(M·N)=lnM+lnN
    Quotient Property of Logarithms logaMN=logaM−logaNlogaMN=logaM−logaN lnMN=lnM−lnNlnMN=lnM−lnN
    Power Property of Logarithms logaMp=plogaMlogaMp=plogaM lnMp=plnMlnMp=plnM
  • Change-of-Base Formula
     For any logarithmic bases a and b, and M>0,M>0,
    logaM=logbMlogbalogaM=logMlogalogaM=lnMlna new basebnew base 10new baseelogaM=logbMlogbalogaM=logMlogalogaM=lnMlna new basebnew base 10new basee

10.5 Solve Exponential and Logarithmic Equations

  • One-to-One Property of Logarithmic Equations: For M>0,N>0,a>0,M>0,N>0,a>0, and a≠1a≠1 is any real number:
    IflogaM=logaN,thenM=N.IflogaM=logaN,thenM=N.
  • Compound Interest:
     For a principal, P, invested at an interest rate, r, for t years, the new balance, A, is:
    A=P(1+rn)ntwhen compoundedntimes a year. A=Pertwhen compounded continuously.A=P(1+rn)ntwhen compoundedntimes a year. A=Pertwhen compounded continuously.
  • Exponential Growth and Decay: For an original amount, A0A0 that grows or decays at a rate, r, for a certain time t, the final amount, A, is A=A0ert.A=A0ert.
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