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Precalculus

Review Exercises

PrecalculusReview Exercises
Table of contents
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Functions
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  14. A | Basic Functions and Identities
  15. 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
  16. Index

Review Exercises

Functions and Function Notation

For the following exercises, determine whether the relation is a function.

1.

{ (a,b),(c,d),(e,d) } { (a,b),(c,d),(e,d) }

2.

{ (5,2),(6,1),(6,2),(4,8) } { (5,2),(6,1),(6,2),(4,8) }

3.

y 2 +4=x, y 2 +4=x, for x x the independent variable and y y the dependent variable

4.

Is the graph in Figure 1 a function?

Graph of a parabola.
Figure 1

For the following exercises, evaluate the function at the indicated values: f(−3);f(2);f(−a);−f(a);f(a+h). f(−3);f(2);f(−a);−f(a);f(a+h).

5.

f(x)=−2 x 2 +3x f(x)=−2 x 2 +3x

6.

f(x)=2| 3x−1 | f(x)=2| 3x−1 |

For the following exercises, determine whether the functions are one-to-one.

7.

f(x)=−3x+5 f(x)=−3x+5

8.

f(x)=| x−3 | f(x)=| x−3 |

For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.

9.
Graph of a cubic function.
10.
Graph of a relation.
11.
Graph of a relation.

For the following exercises, graph the functions.

12.

f(x)=| x+1 | f(x)=| x+1 |

13.

f(x)= x 2 −2 f(x)= x 2 −2

For the following exercises, use Figure 2 to approximate the values.

Graph of a parabola.
Figure 2
14.

f(2) f(2)

15.

f(−2) f(−2)

16.

If f(x)=−2, f(x)=−2, then solve for x. x.

17.

If f(x)=1, f(x)=1, then solve for x. x.

For the following exercises, use the function h(t)=−16 t 2 +80t h(t)=−16 t 2 +80t to find the values.

18.

h(2)−h(1) 2−1 h(2)−h(1) 2−1

19.

h(a)−h(1) a−1 h(a)−h(1) a−1

Domain and Range

For the following exercises, find the domain of each function, expressing answers using interval notation.

20.

f(x)= 2 3x+2 f(x)= 2 3x+2

21.

f(x)= x−3 x 2 −4x−12 f(x)= x−3 x 2 −4x−12

22.

f(x)= x−6 x−4 f(x)= x−6 x−4

23.

Graph this piecewise function: f(x)={ x+1        x<−2 −2x−3   x≥−2 f(x)={ x+1        x<−2 −2x−3   x≥−2

Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from x=1 to x=2. x=1 to x=2.

24.

f(x)=4x−3 f(x)=4x−3

25.

f(x)=10 x 2 +x f(x)=10 x 2 +x

26.

f(x)=− 2 x 2 f(x)=− 2 x 2

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

27.
Graph of a parabola.
28.
Graph of a cubic function.
29.
Graph of a function.
30.

Find the local minimum of the function graphed in Exercise 1.27.

31.

Find the local extrema for the function graphed in Exercise 1.28.

32.

For the graph in Figure 3, the domain of the function is [ −3,3 ]. [ −3,3 ]. The range is [ −10,10 ]. [ −10,10 ]. Find the absolute minimum of the function on this interval.

33.

Find the absolute maximum of the function graphed in Figure 3.

Graph of a cubic function.
Figure 3
Composition of Functions

For the following exercises, find (f∘g)(x) (f∘g)(x) and (g∘f)(x) (g∘f)(x) for each pair of functions.

34.

f(x)=4−x,g(x)=−4x f(x)=4−x,g(x)=−4x

35.

f(x)=3x+2,g(x)=5−6x f(x)=3x+2,g(x)=5−6x

36.

f(x)= x 2 +2x,g(x)=5x+1 f(x)= x 2 +2x,g(x)=5x+1

37.

f(x)= x+2 ,g(x)= 1 x f(x)= x+2 ,g(x)= 1 x

38.

f(x)= x+3 2 ,g(x)= 1−x f(x)= x+3 2 ,g(x)= 1−x

For the following exercises, find ( f∘g ) ( f∘g ) and the domain for ( f∘g )(x) ( f∘g )(x) for each pair of functions.

39.

f(x)= x+1 x+4 ,g(x)= 1 x f(x)= x+1 x+4 ,g(x)= 1 x

40.

f(x)= 1 x+3 ,g(x)= 1 x−9 f(x)= 1 x+3 ,g(x)= 1 x−9

41.

f(x)= 1 x ,g(x)= x f(x)= 1 x ,g(x)= x

42.

f(x)= 1 x 2 −1 ,g(x)= x+1 f(x)= 1 x 2 −1 ,g(x)= x+1

For the following exercises, express each function H H as a composition of two functions f f and g g where H(x)=(f∘g)(x). H(x)=(f∘g)(x).

43.

H(x)= 2x−1 3x+4 H(x)= 2x−1 3x+4

44.

H(x)= 1 (3 x 2 −4) −3 H(x)= 1 (3 x 2 −4) −3

Transformation of Functions

For the following exercises, sketch a graph of the given function.

45.

f(x)= (x−3) 2 f(x)= (x−3) 2

46.

f(x)= (x+4) 3 f(x)= (x+4) 3

47.

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

48.

f(x)=− x 3 f(x)=− x 3

49.

f(x)= −x 3 f(x)= −x 3

50.

f(x)=5 −x −4 f(x)=5 −x −4

51.

f(x)=4[ | x−2 |−6 ] f(x)=4[ | x−2 |−6 ]

52.

f(x)=− (x+2) 2 −1 f(x)=− (x+2) 2 −1

For the following exercises, sketch the graph of the function g g if the graph of the function f f is shown in Figure 4.

Graph of f(x)
Figure 4
53.

g(x)=f(x−1) g(x)=f(x−1)

54.

g(x)=3f(x) g(x)=3f(x)

For the following exercises, write the equation for the standard function represented by each of the graphs below.

55.
Graph of an absolute function.
56.
Graph of a half circle.

For the following exercises, determine whether each function below is even, odd, or neither.

57.

f(x)=3 x 4 f(x)=3 x 4

58.

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

59.

h(x)= 1 x +3x h(x)= 1 x +3x

For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.

60.
Graph of a parabola.
61.
Graph of a parabola.
62.
Graph of a cubic function.
Absolute Value Functions

For the following exercises, write an equation for the transformation of f(x)=| x |. f(x)=| x |.

63.
Graph of f(x).
64.
Graph of f(x).
65.
Graph of f(x).

For the following exercises, graph the absolute value function.

66.

f(x)=| x−5 | f(x)=| x−5 |

67.

f(x)=−| x−3 | f(x)=−| x−3 |

68.

f(x)=| 2x−4 | f(x)=| 2x−4 |

For the following exercises, solve the absolute value equation.

69.

| x+4 |=18 | x+4 |=18

70.

| 1 3 x+5 |=| 3 4 x−2 | | 1 3 x+5 |=| 3 4 x−2 |

For the following exercises, solve the inequality and express the solution using interval notation.

71.

| 3x−2 |<7 | 3x−2 |<7

72.

| 1 3 x−2 |≤7 | 1 3 x−2 |≤7

Inverse Functions

For the following exercises, find f −1 (x) f −1 (x) for each function.

73.

f(x)=9+10x f(x)=9+10x

74.

f(x)= x x+2 f(x)= x x+2

For the following exercise, find a domain on which the function f f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f f restricted to that domain.

75.

f(x)= x 2 +1 f(x)= x 2 +1

76.

Given f( x )= x 3 −5 f( x )= x 3 −5 and g(x)= x+5 3 : g(x)= x+5 3 :

  1. Find f(g(x)) f(g(x)) and g(f(x)). g(f(x)).
  2. What does the answer tell us about the relationship between f(x) f(x) and g(x)? g(x)?

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

77.

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

78.

f(x)=−3 x 2 +x f(x)=−3 x 2 +x

79.

If f( 5 )=2, f( 5 )=2, find f −1 (2). f −1 (2).

80.

If f( 1 )=4, f( 1 )=4, find f −1 (4). f −1 (4).

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