Skip to Content
OpenStax Logo
Buy book
  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. 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
    13. Chapter 13
  17. Index

Try It

5.1 Quadratic Functions

1.

The path passes through the origin and has vertex at ( 4, 7 ), ( 4, 7 ), so h(x)= 7 16 (x+4) 2 +7. h(x)= 7 16 (x+4) 2 +7. To make the shot, h( 7.5 ) h( 7.5 ) would need to be about 4 but h(7.5)1.64; h(7.5)1.64; he doesn’t make it.

2.

g(x)= x 2 6x+13 g(x)= x 2 6x+13 in general form; g(x)= (x3) 2 +4 g(x)= (x3) 2 +4 in standard form

3.

The domain is all real numbers. The range is f(x) 8 11 , f(x) 8 11 , or [ 8 11 , ). [ 8 11 , ).

4.

y-intercept at (0, 13), No x- x- intercepts

5.

3 seconds 256 feet 7 seconds

5.2 Power Functions and Polynomial Functions

1.

f(x) f(x) is a power function because it can be written as f(x)=8 x 5 . f(x)=8 x 5 . The other functions are not power functions.

2.

As x x approaches positive or negative infinity, f( x ) f( x ) decreases without bound: as x±, f(x) x±, f(x) because of the negative coefficient.

3.

The degree is 6. The leading term is x 6 . x 6 . The leading coefficient is 1. 1.

4.

As x, f(x); as x, f(x). x, f(x); as x, f(x). It has the shape of an even degree power function with a negative coefficient.

5.

The leading term is 0.2 x 3 , 0.2 x 3 , so it is a degree 3 polynomial. As x x approaches positive infinity, f( x ) f( x ) increases without bound; as x x approaches negative infinity, f( x ) f( x ) decreases without bound.

6.

y-intercept (0,0); (0,0); x-intercepts (0,0),(2,0), (0,0),(2,0), and (5,0) (5,0)

7.

There are at most 12 x- x- intercepts and at most 11 turning points.

8.

The end behavior indicates an odd-degree polynomial function; there are 3 x- x- intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.

9.

The x- x- intercepts are (2,0),(1,0), (2,0),(1,0), and (5,0), (5,0), the y-intercept is (0,2), (0,2), and the graph has at most 2 turning points.

5.3 Graphs of Polynomial Functions

1.

y-intercept (0,0); (0,0); x-intercepts (0,0),(5,0),(2,0), (0,0),(5,0),(2,0), and (3,0) (3,0)

2.

The graph has a zero of –5 with multiplicity 3, a zero of -1 with multiplicity 2, and a zero of 3 with multiplicity 4.

3.
Graph of f(x)=(1/4)x(x-1)^4(x+3)^3.
4.

Because f f is a polynomial function and since f(1) f(1) is negative and f(2) f(2) is positive, there is at least one real zero between x=1 x=1 and x=2. x=2.

5.

f(x)= 1 8 (x2) 3 (x+1) 2 (x4) f(x)= 1 8 (x2) 3 (x+1) 2 (x4)

6.

The minimum occurs at approximately the point (0,6.5), (0,6.5), and the maximum occurs at approximately the point (3.5,7). (3.5,7).

5.4 Dividing Polynomials

1.

4 x 2 8x+15 78 4x+5 4 x 2 8x+15 78 4x+5

2.

3 x 3 3 x 2 +21x150+ 1,090 x+7 3 x 3 3 x 2 +21x150+ 1,090 x+7

3.

3 x 2 4x+1 3 x 2 4x+1

5.5 Zeros of Polynomial Functions

1.

f(3)=412 f(3)=412

2.

The zeros are 2, –2, and –4.

3.

There are no rational zeros.

4.

The zeros are –4,  1 2 , and 1. –4,  1 2 , and 1.

5.

f(x)= 1 2 x 3 + 5 2 x 2 2x+10 f(x)= 1 2 x 3 + 5 2 x 2 2x+10

6.

There must be 4, 2, or 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros.

7.

3 meters by 4 meters by 7 meters

5.6 Rational Functions

1.

End behavior: as x±, f(x)0; x±, f(x)0; Local behavior: as x0, f(x) x0, f(x) (there are no x- or y-intercepts)

2.
Graph of f(x)=1/(x-3)^2-4 with its vertical asymptote at x=3 and its horizontal asymptote at y=-4.

The function and the asymptotes are shifted 3 units right and 4 units down. As x3,f(x), x3,f(x), and as x±,f(x)4. x±,f(x)4.

The function is f(x)= 1 (x3) 2 4. f(x)= 1 (x3) 2 4.

3.

12 11 12 11

4.

The domain is all real numbers except x=1 x=1 and x=5. x=5.

5.

Removable discontinuity at x=5. x=5. Vertical asymptotes: x=0, x=1. x=0, x=1.

6.

Vertical asymptotes at x=2 x=2 and x=3; x=3; horizontal asymptote at y=4. y=4.

7.

For the transformed reciprocal squared function, we find the rational form. f(x)= 1 (x3) 2 4= 14 (x3) 2 (x3) 2 = 14( x 2 6x+9) (x3)(x3) = 4 x 2 +24x35 x 2 6x+9 f(x)= 1 (x3) 2 4= 14 (x3) 2 (x3) 2 = 14( x 2 6x+9) (x3)(x3) = 4 x 2 +24x35 x 2 6x+9

Because the numerator is the same degree as the denominator we know that as x±, f(x)4; so y=4 x±, f(x)4; so y=4 is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is x=3, x=3, because as x3,f(x). x3,f(x). We then set the numerator equal to 0 and find the x-intercepts are at (2.5,0) (2.5,0) and (3.5,0). (3.5,0). Finally, we evaluate the function at 0 and find the y-intercept to be at ( 0, 35 9 ). ( 0, 35 9 ).

8.

Horizontal asymptote at y= 1 2 . y= 1 2 . Vertical asymptotes at x=1 and x=3. x=1 and x=3. y-intercept at ( 0, 4 3 . ) ( 0, 4 3 . )

x-intercepts at (2,0)  and (2,0). (2,0)  and (2,0). (2,0) (2,0) is a zero with multiplicity 2, and the graph bounces off the x-axis at this point. (2,0) (2,0) is a single zero and the graph crosses the axis at this point.

Graph of f(x)=(x+2)^2(x-2)/2(x-1)^2(x-3) with its vertical and horizontal asymptotes.

5.7 Inverses and Radical Functions

1.

f 1 ( f( x ) )= f 1 ( x+5 3 )=3( x+5 3 )5=( x5 )+5=x f 1 ( f( x ) )= f 1 ( x+5 3 )=3( x+5 3 )5=( x5 )+5=x and f( f 1 ( x ) )=f( 3x5 )= ( 3x5 )+5 3 = 3x 3 =x f( f 1 ( x ) )=f( 3x5 )= ( 3x5 )+5 3 = 3x 3 =x

2.

f 1 (x)= x 3 4 f 1 (x)= x 3 4

3.

f 1 (x)= x1 f 1 (x)= x1

4.

f 1 (x)= x 2 3 2 ,x0 f 1 (x)= x 2 3 2 ,x0

5.

f 1 (x)= 2x+3 x1 f 1 (x)= 2x+3 x1

5.8 Modeling Using Variation

1.

128 3 128 3

2.

9 2 9 2

3.

x=20 x=20

5.1 Section Exercises

1.

When written in that form, the vertex can be easily identified.

3.

If a=0 a=0 then the function becomes a linear function.

5.

If possible, we can use factoring. Otherwise, we can use the quadratic formula.

7.

g(x)= (x+1) 2 4, g(x)= (x+1) 2 4, Vertex ( 1,4 ) ( 1,4 )

9.

f(x)= ( x+ 5 2 ) 2 33 4 , f(x)= ( x+ 5 2 ) 2 33 4 , Vertex ( 5 2 , 33 4 ) ( 5 2 , 33 4 )

11.

f(x)=3 (x1) 2 12, f(x)=3 (x1) 2 12, Vertex (1,12) (1,12)

13.

f(x)=3 ( x 5 6 ) 2 37 12 , f(x)=3 ( x 5 6 ) 2 37 12 , Vertex ( 5 6 , 37 12 ) ( 5 6 , 37 12 )

15.

Minimum is 17 2 17 2 and occurs at 5 2 . 5 2 . Axis of symmetry is x= 5 2 . x= 5 2 .

17.

Minimum is 17 16 17 16 and occurs at 1 8 . 1 8 . Axis of symmetry is x= 1 8 . x= 1 8 .

19.

Minimum is 7 2 7 2 and occurs at −3. −3. Axis of symmetry is x=−3. x=−3.

21.

Domain is ( , ). ( , ). Range is [2,). [2,).

23.

Domain is ( , ). ( , ). Range is [−5,). [−5,).

25.

Domain is ( , ). ( , ). Range is [−12,). [−12,).

27.

f(x)= x 2 +4x+3 f(x)= x 2 +4x+3

29.

f(x)= x 2 -4x+7 f(x)= x 2 -4x+7

31.

f(x)= -149 x 2 +649x +8949 f(x)= -149 x 2 +649x +8949

33.

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

35.

Vertex: (3, −10), axis of symmetry: x = 3, intercepts: (3+10,0)(3+10,0) and (3-10,0)(3-10,0)

37.

Vertex: (32,-12) (32,-12) , axis of symmetry: x=32 x=32 , intercept: ( 3+23 2 , 0) ( 3+23 2 , 0) and ( 3-23 2 , 0) ( 3-23 2 , 0)

39.
Graph of f(x)=4x^2-12x-3
41.

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

43.

f(x)=-3 x 2 6x1 f(x)=-3 x 2 6x1

45.

f(x)=-14 x 2 x+2 f(x)=-14 x 2 x+2

47.

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

49.

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

50.

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

51.

The value stretches or compresses the width of the graph. The greater the value, the narrower the graph.

53.

The graph is shifted to the right or left (a horizontal shift).

55.

The suspension bridge has 1,000 feet distance from the center.

57.

Domain is (,). (,). Range is (-,2]. (-,2].

59.

Domain: (-,) (-,) ; range: [100,) [100,)

61.

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

63.

f(x)=- x 2 2 f(x)=- x 2 2

65.

f(x)=3 x 2 +6x-15 f(x)=3 x 2 +6x-15

67.

75 feet by 50 feet

69.

3 and 3; product is 9

71.

The revenue reaches the maximum value when 1800 thousand phones are produced.

73.

2.449 seconds

75.

41 trees per acre

5.2 Section Exercises

1.

The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

3.

As x x decreases without bound, so does f( x ). f( x ). As x x increases without bound, so does f( x ). f( x ).

5.

The polynomial function is of even degree and leading coefficient is negative.

7.

Power function

9.

Neither

11.

Neither

13.

Degree = 2, Coefficient = –2

15.

Degree =4, Coefficient = –2

17.

Asx,f(x),asx,f(x) Asx,f(x),asx,f(x)

19.

Asx,f(x),asx,f(x) Asx,f(x),asx,f(x)

21.

Asx,f(x),asx,f(x) Asx,f(x),asx,f(x)

23.

Asx,f(x),asx,f(x) Asx,f(x),asx,f(x)

25.

y-intercept is (0,12), (0,12), t-intercepts are (1,0);(2,0);and (3,0). (1,0);(2,0);and (3,0).

27.

y-intercept is (0,16). (0,16). x-intercepts are (2,0) (2,0) and (2,0). (2,0).

29.

y-intercept is (0,0). (0,0). x-intercepts are (0,0),(4,0), (0,0),(4,0), and ( 2, 0 ). ( 2, 0 ).

31.

3

33.

5

35.

3

37.

5

39.

Yes. Number of turning points is 2. Least possible degree is 3.

41.

Yes. Number of turning points is 1. Least possible degree is 2.

43.

Yes. Number of turning points is 0. Least possible degree is 1.

45.

Yes. Number of turning points is 0. Least possible degree is 1.

47.

x x f( x ) f( x )
10 9,500
100 99,950,000
–10 9,500
–100 99,950,000

asx,f(x),asx,f(x) asx,f(x),asx,f(x)

49.

x x f( x ) f( x )
10 –504
100 –941,094
–10 1,716
–100 1,061,106

asx,f(x),asx,f(x) asx,f(x),asx,f(x)

51.
Graph of f(x)=x^3(x-2).

The y- y- intercept is ( 0, 0 ). ( 0, 0 ). The x- x- intercepts are ( 0, 0 ), ( 2, 0 ). ( 0, 0 ), ( 2, 0 ). Asx,f(x),asx,f(x) Asx,f(x),asx,f(x)

53.
Graph of f(x)=x(14-2x)(10-2x).

The y- y- intercept is ( 0,0 ) ( 0,0 ) . The x- x- intercepts are ( 0, 0 ), ( 5, 0 ), ( 7, 0 ). ( 0, 0 ), ( 5, 0 ), ( 7, 0 ). Asx,f(x),asx,f(x) Asx,f(x),asx,f(x)

55.

The y- y- intercept is ( 0, 0 ). ( 0, 0 ). The x- x- intercept is ( 4, 0 ), ( 0, 0 ), ( 4, 0 ). ( 4, 0 ), ( 0, 0 ), ( 4, 0 ). Asx,f(x),asx,f(x) Asx,f(x),asx,f(x)

57.
Graph of f(x)=x^3-27.

The y- y- intercept is ( 0, 81 ). ( 0, 81 ). The x- x- intercept are ( 3, 0 ), ( 3, 0 ). ( 3, 0 ), ( 3, 0 ). Asx,f(x),asx,f(x) Asx,f(x),asx,f(x)

59.
Graph of f(x)=-x^3+x^2+2x.

The y- y- intercept is ( 0, 0 ). ( 0, 0 ). The x- x- intercepts are ( 3, 0 ), ( 0, 0 ), ( 5, 0 ). ( 3, 0 ), ( 0, 0 ), ( 5, 0 ). Asx,f(x),asx,f(x) Asx,f(x),asx,f(x)

61.

f(x)= x 2 4 f(x)= x 2 4

63.

f(x)= x 3 4 x 2 +4x f(x)= x 3 4 x 2 +4x

65.

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

67.

V(m)=8 m 3 +36 m 2 +54m+27 V(m)=8 m 3 +36 m 2 +54m+27

69.

V(x)=4 x 3 32 x 2 +64x V(x)=4 x 3 32 x 2 +64x

5.3 Section Exercises

1.

The x- x- intercept is where the graph of the function crosses the x- x- axis, and the zero of the function is the input value for which f(x)=0. f(x)=0.

3.

If we evaluate the function at a a and at b b and the sign of the function value changes, then we know a zero exists between a a and b. b.

5.

There will be a factor raised to an even power.

7.

(2,0),(3,0),(5,0) (2,0),(3,0),(5,0)

9.

(3,0),(1,0),(0,0) (3,0),(1,0),(0,0)

11.

( 0,0 ), ( 5,0 ), ( 2,0 ) ( 0,0 ), ( 5,0 ), ( 2,0 )

13.

( 0,0 ), ( 5,0 ), ( 4,0 ) ( 0,0 ), ( 5,0 ), ( 4,0 )

15.

( 2,0 ), ( 2,0 ), ( 1,0 ) ( 2,0 ), ( 2,0 ), ( 1,0 )

17.

(2,0),(2,0),( 1 2 ,0 ) (2,0),(2,0),( 1 2 ,0 )

19.

( 1,0 ), ( 1,0 ) ( 1,0 ), ( 1,0 )

21.

(0,0),( 3 ,0),( 3 ,0) (0,0),( 3 ,0),( 3 ,0)

23.

( 0,0 ), ( 1,0 )( 1,0 ), ( 2,0 ), ( 2,0 ) ( 0,0 ), ( 1,0 )( 1,0 ), ( 2,0 ), ( 2,0 )

25.

f( 2 )=10 f( 2 )=10 and f( 4 )=28. f( 4 )=28. Sign change confirms.

27.

f( 1 )=3 f( 1 )=3 and f( 3 )=77. f( 3 )=77. Sign change confirms.

29.

f( 0.01 )=1.000001 f( 0.01 )=1.000001 and f( 0.1 )=7.999. f( 0.1 )=7.999. Sign change confirms.

31.

0 with multiplicity 2, 3 2 3 2 with multiplicity 5, 4 with multiplicity 2

33.

0 with multiplicity 2, –2 with multiplicity 2

35.

2 3 withmultiplicity5,5withmultiplicity2 2 3 withmultiplicity5,5withmultiplicity2

37.

0withmultiplicity4,2withmultiplicity1,1withmultiplicity1 0withmultiplicity4,2withmultiplicity1,1withmultiplicity1

39.

3 2 3 2 with multiplicity 2, 0 with multiplicity 3

41.

0withmultiplicity6, 2 3 withmultiplicity2 0withmultiplicity6, 2 3 withmultiplicity2

43.

x-intercepts, ( 1, 0 ) ( 1, 0 ) with multiplicity 2, ( 4, 0 ) ( 4, 0 ) with multiplicity 1, y- y- intercept ( 0, 4 ). ( 0, 4 ). As x,f(x),asx,f(x). x,f(x),asx,f(x).

Graph of g(x)=(x+4)(x-1)^2.
45.

x-intercepts (3,0) (3,0) with multiplicity 3, (2,0) (2,0) with multiplicity 2, y- y- intercept (0,108). (0,108). As x,f(x),asx,f(x). x,f(x),asx,f(x).

Graph of k(x)=(x-3)^3(x-2)^2.
47.

x-intercepts ( 0, 0 ), ( 2, 0 ), ( 4,0 ) ( 0, 0 ), ( 2, 0 ), ( 4,0 ) with multiplicity 1, y- y- intercept (0, 0). (0, 0). As x,f(x),asx,f(x). x,f(x),asx,f(x).

Graph of n(x)=-3x(x+2)(x-4).
49.

f(x)= 2 9 (x3)(x+1)(x+3) f(x)= 2 9 (x3)(x+1)(x+3)

51.

f(x)= 1 4 (x+2) 2 (x3) f(x)= 1 4 (x+2) 2 (x3)

53.

–4, –2, 1, 3 with multiplicity 1

55.

–2, 3 each with multiplicity 2

57.

f(x)= 2 3 (x+2)(x1)(x3) f(x)= 2 3 (x+2)(x1)(x3)

59.

f(x)= 1 3 (x3) 2 (x1) 2 (x+3) f(x)= 1 3 (x3) 2 (x1) 2 (x+3)

61.

f(x)=−15 (x1) 2 (x3) 3 f(x)=−15 (x1) 2 (x3) 3

63.

f(x)=2( x+3 )( x+2 )( x1 ) f(x)=2( x+3 )( x+2 )( x1 )

65.

f(x)= 3 2 ( 2x1 ) 2 ( x6 )( x+2 ) f(x)= 3 2 ( 2x1 ) 2 ( x6 )( x+2 )

67.

local max ( .58, –.62 ), ( .58, –.62 ), local min ( .58, –1.38 ) ( .58, –1.38 )

69.

global min ( .63, –.47 ) ( .63, –.47 )

71.

global min (.75, .89) (.75, .89)

73.

f(x)= (x500) 2 (x+200) f(x)= (x500) 2 (x+200)

75.

f(x)=4 x 3 36 x 2 +80x f(x)=4 x 3 36 x 2 +80x

77.

f(x)=4 x 3 36 x 2 +60x+100 f(x)=4 x 3 36 x 2 +60x+100

79.

f(x)=9π( x 3 +5 x 2 +8x+4) f(x)=9π( x 3 +5 x 2 +8x+4)

5.4 Section Exercises

1.

The binomial is a factor of the polynomial.

3.

x+6+ 5 x-1 ,quotient:x+6,remainder:5 x+6+ 5 x-1 ,quotient:x+6,remainder:5

5.

3x+2,quotient: 3x+2,remainder: 0 3x+2,quotient: 3x+2,remainder: 0

7.

x5,quotient:x5,remainder:0 x5,quotient:x5,remainder:0

9.

2x7+ 16 x+2 ,quotient:2x7,remainder:16 2x7+ 16 x+2 ,quotient:2x7,remainder:16

11.

x2+ 6 3x+1 ,quotient:x2,remainder:6 x2+ 6 3x+1 ,quotient:x2,remainder:6

13.

2 x 2 3x+5,quotient:2 x 2 3x+5,remainder:0 2 x 2 3x+5,quotient:2 x 2 3x+5,remainder:0

15.

2 x 2 +2x+1+ 10 x4 2 x 2 +2x+1+ 10 x4

17.

2 x 2 7x+1 2 2x+1 2 x 2 7x+1 2 2x+1

19.

3 x 2 11x+34 106 x+3 3 x 2 11x+34 106 x+3

21.

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

23.

4 x 2 21x+84 323 x+4 4 x 2 21x+84 323 x+4

25.

x 2 14x+49 x 2 14x+49

27.

3 x 2 +x+ 2 3x1 3 x 2 +x+ 2 3x1

29.

x 3 3x+1 x 3 3x+1

31.

x 3 x 2 +2 x 3 x 2 +2

33.

x 3 6 x 2 +12x8 x 3 6 x 2 +12x8

35.

x 3 9 x 2 +27x27 x 3 9 x 2 +27x27

37.

2 x 3 2x+2 2 x 3 2x+2

39.

Yes ( x2 )(3 x 3 5) ( x2 )(3 x 3 5)

41.

Yes ( x2 )(4 x 3 +8 x 2 +x+2) ( x2 )(4 x 3 +8 x 2 +x+2)

43.

No

45.

(x1)( x 2 +2x+4) (x1)( x 2 +2x+4)

47.

(x5)( x 2 +x+1) (x5)( x 2 +x+1)

49.

Quotient:4 x 2 +8x+16,remainder:1 Quotient:4 x 2 +8x+16,remainder:1

51.

Quotient:3 x 2 +3x+5,remainder:0 Quotient:3 x 2 +3x+5,remainder:0

53.

Quotient: x 3 2 x 2 +4x8,remainder:6 Quotient: x 3 2 x 2 +4x8,remainder:6

55.

x 6 x 5 + x 4 x 3 + x 2 x+1 x 6 x 5 + x 4 x 3 + x 2 x+1

57.

x 3 x 2 +x1+ 1 x+1 x 3 x 2 +x1+ 1 x+1

59.

1+ 1+i xi 1+ 1+i xi

61.

1+ 1i x+i 1+ 1i x+i

63.

x 2 ix1+ 1i xi x 2 ix1+ 1i xi

65.

2 x 2 +3 2 x 2 +3

67.

2x+3 2x+3

69.

x+2 x+2

71.

x3 x3

73.

3 x 2 2 3 x 2 2

5.5 Section Exercises

1.

The theorem can be used to evaluate a polynomial.

3.

Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.

5.

Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

7.

106 106

9.

0 0

11.

255 255

13.

1 1

15.

2, 1,  1 2 2, 1,  1 2

17.

2 2

19.

3 3

21.

5 2 ,  6 ,  6 5 2 ,  6 ,  6

23.

2, 4,  3 2 2, 4,  3 2

25.

4, 4, 5 4, 4, 5

27.

5, 3,  1 2 5, 3,  1 2

29.

1 2 ,  1+ 5 2 ,  1 5 2 1 2 ,  1+ 5 2 ,  1 5 2

31.

3 2 3 2

33.

2, 3, 1, 2 2, 3, 1, 2

35.

1 2 ,  1 2 , 2, 3 1 2 ,  1 2 , 2, 3

37.

1, 1,  5 ,  5 1, 1,  5 ,  5

39.

3 4 ,  1 2 3 4 ,  1 2

41.

2, 3+2i, 32i 2, 3+2i, 32i

43.

2 3 , 1+2i, 12i 2 3 , 1+2i, 12i

45.

1 2 , 1+4i, 14i 1 2 , 1+4i, 14i

47.

1 positive, 1 negative

Graph of f(x)=x^4-x^2-1.
49.

3 or 1 positive, 0 negative

Graph of f(x)=x^3-2x^2+x-1.
51.

0 positive, 3 or 1 negative

Graph of f(x)=2x^3+37x^2+200x+300.
53.

2 or 0 positive, 2 or 0 negative

Graph of f(x)=2x^4-5x^3-5x^2+5x+3.
55.

2 or 0 positive, 2 or 0 negative

Graph of f(x)=10x^4-21x^2+11.
57.

±5, ±1, ± 5 2 ±5, ±1, ± 5 2

59.

±1, ± 1 2 , ± 1 3 , ± 1 6 ±1, ± 1 2 , ± 1 3 , ± 1 6

61.

1,  1 2 ,  1 3 1,  1 2 ,  1 3

63.

2,  1 4 ,  3 2 2,  1 4 ,  3 2

65.

5 4 5 4

67.

f(x)= 4 9 ( x 3 + x 2 x1 ) f(x)= 4 9 ( x 3 + x 2 x1 )

69.

f(x)= 1 5 ( 4 x 3 x ) f(x)= 1 5 ( 4 x 3 x )

71.

8 by 4 by 6 inches

73.

5.5 by 4.5 by 3.5 inches

75.

8 by 5 by 3 inches

77.

Radius = 6 meters, Height = 2 meters

79.

Radius = 2.5 meters, Height = 4.5 meters

5.6 Section Exercises

1.

The rational function will be represented by a quotient of polynomial functions.

3.

The numerator and denominator must have a common factor.

5.

Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator.

7.

All reals x1, 1 All reals x1, 1

9.

All reals x1, 2, 1, 2 All reals x1, 2, 1, 2

11.

V.A. at x= 2 5 ; x= 2 5 ; H.A. at y=0; y=0; Domain is all reals x 2 5 x 2 5

13.

V.A. at x=4, 9; x=4, 9; H.A. at y=0; y=0; Domain is all reals x4, 9 x4, 9

15.

V.A. at x=0, 4, 4; x=0, 4, 4; H.A. at y=0; y=0; Domain is all reals x0,4, 4 x0,4, 4

17.

V.A. at x=5; x=5; H.A. at y=0; y=0; Domain is all reals x5,5 x5,5

19.

V.A. at x= 1 3 ; x= 1 3 ; H.A. at y= 2 3 ; y= 2 3 ; Domain is all reals x 1 3 . x 1 3 .

21.

none

23.

x-intercepts none, y-intercept ( 0, 1 4 ) x-intercepts none, y-intercept ( 0, 1 4 )

25.

Local behavior: x 1 2 + ,f(x),x 1 2 ,f(x) x 1 2 + ,f(x),x 1 2 ,f(x)

End behavior: x±,f(x) 1 2 x±,f(x) 1 2

27.

Local behavior: x 6 + ,f(x),x 6 ,f(x), x 6 + ,f(x),x 6 ,f(x), End behavior: x±,f(x)2 x±,f(x)2

29.

Local behavior: x 1 3 + ,f(x),x 1 3 , x 1 3 + ,f(x),x 1 3 , f(x),x 5 2 ,f(x),x 5 2 + ,f(x) f(x),x 5 2 ,f(x),x 5 2 + ,f(x)

End behavior: x±,f(x) 1 3 x±,f(x) 1 3

31.

y=2x+4 y=2x+4

33.

y=2x y=2x

35.

V.A. x=0,H.A. y=2 V.A. x=0,H.A. y=2

Graph of a rational function.
37.

V.A. x=2, H.A. y=0 V.A. x=2, H.A. y=0

Graph of a rational function.
39.

V.A. x=4, H.A. y=2;( 3 2 ,0 );( 0, 3 4 ) V.A. x=4, H.A. y=2;( 3 2 ,0 );( 0, 3 4 )

Graph of p(x)=(2x-3)/(x+4) with its vertical asymptote at x=-4 and horizontal asymptote at y=2.
41.

V.A. x=2, H.A. y=0, (0,1) V.A. x=2, H.A. y=0, (0,1)

Graph of s(x)=4/(x-2)^2 with its vertical asymptote at x=2 and horizontal asymptote at y=0.
43.

V.A. x=4, x= 4 3 , H.A. y=1;(5,0);( 1 3 ,0 );( 0, 5 16 ) V.A. x=4, x= 4 3 , H.A. y=1;(5,0);( 1 3 ,0 );( 0, 5 16 )

Graph of f(x)=(3x^2-14x-5)/(3x^2+8x-16) with its vertical asymptotes at x=-4 and x=4/3 and horizontal asymptote at y=1.
45.

V.A. x=1, H.A. y=1;( 3,0 );( 0,3 ) V.A. x=1, H.A. y=1;( 3,0 );( 0,3 )

Graph of a(x)=(x^2+2x-3)/(x^2-1) with its vertical asymptote at x=-1 and horizontal asymptote at y=1.
47.

V.A. x=4, S.A. y=2x+9;( 1,0 );( 1 2 ,0 );( 0, 1 4 ) V.A. x=4, S.A. y=2x+9;( 1,0 );( 1 2 ,0 );( 0, 1 4 )

Graph of h(x)=(2x^2+x-1)/(x-1) with its vertical asymptote at x=4 and slant asymptote at y=2x+9.
49.

V.A. x=2, x=4, H.A. y=1,( 1,0 );( 5,0 );( 3,0 );( 0, 15 16 ) V.A. x=2, x=4, H.A. y=1,( 1,0 );( 5,0 );( 3,0 );( 0, 15 16 )

Graph of w(x)=(x-1)(x+3)(x-5)/(x+2)^2(x-4) with its vertical asymptotes at x=-2 and x=4 and horizontal asymptote at y=1.
51.

y=50 x 2 x2 x 2 25 y=50 x 2 x2 x 2 25

53.

y=7 x 2 +2x24 x 2 +9x+20 y=7 x 2 +2x24 x 2 +9x+20

55.

y= 1 2 x 2 4x+4 x+1 y= 1 2 x 2 4x+4 x+1

57.

y=4 x3 x 2 x12 y=4 x3 x 2 x12

59.

y= 27(x2) (x+3) (x3)2 y= 27(x2) (x+3) (x3)2

61.

y= 1 3 x 2 +x6 x1 y= 1 3 x 2 +x6 x1

63.

y=6 (x1) 2 (x+3) (x2) 2 y=6 (x1) 2 (x+3) (x2) 2

65.

x x 2.01 2.001 2.0001 1.99 1.999
y y 100 1,000 10,000 –100 –1,000

x x 10 100 1,000 10,000 100,000
y y .125 .0102 .001 .0001 .00001

Vertical asymptote x=2, x=2, Horizontal asymptote y=0 y=0

67.

x x –4.1 –4.01 –4.001 –3.99 –3.999
y y 82 802 8,002 –798 –7998

x x 10 100 1,000 10,000 100,000
y y 1.4286 1.9331 1.992 1.9992 1.999992

Vertical asymptote x=4, x=4, Horizontal asymptote y=2 y=2

69.

x x –.9 –.99 –.999 –1.1 –1.01
y y 81 9,801 998,001 121 10,201

x x 10 100 1,000 10,000 100,000
y y .82645 .9803 .998 .9998

Vertical asymptote x=1, x=1, Horizontal asymptote y=1 y=1

71.

( 3 2 , ) ( 3 2 , )

Graph of f(x)=4/(2x-3).
73.

(2,1)(4,) (2,1)(4,)

Graph of f(x)=(x+2)/(x-1)(x-4).
75.

( 2,4 ) ( 2,4 )

77.

( 2,5 ) ( 2,5 )

79.

( 1,1 ) ( 1,1 )

81.

C(t)= 8+2t 300+20t C(t)= 8+2t 300+20t

83.

After about 6.12 hours.

85.

A(x)=50 x 2 + 800 x . A(x)=50 x 2 + 800 x . 2 by 2 by 5 feet.

87.

A(x)=π x 2 + 100 x . A(x)=π x 2 + 100 x . Radius = 2.52 meters.

5.7 Section Exercises

1.

It can be too difficult or impossible to solve for x x in terms of y. y.

3.

We will need a restriction on the domain of the answer.

5.

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

7.

f 1 (x)= x+3 1 f 1 (x)= x+3 1

9.

f 1 (x)= 12x f 1 (x)= 12x

11.

f 1 (x)=± x4 2 f 1 (x)=± x4 2

13.

f 1 (x)= x1 3 3 f 1 (x)= x1 3 3

15.

f 1 (x)= 4x 2 3 f 1 (x)= 4x 2 3

17.

f −1 (x)= 3x2 4 ,[ 0, ) f −1 (x)= 3x2 4 ,[ 0, )

19.

f −1 (x)= (x-5)2+8 6 f −1 (x)= (x-5)2+8 6

21.

f −1 (x)= (3-x)2 f −1 (x)=(3-x)2

23.

f −1 (x)= 4x+3 x f −1 (x)= 4x+3 x

25.

f −1 (x)= 7x3 1x f −1 (x)= 7x3 1x

27.

f −1 (x)= 2x-1 5x+5 f −1 (x)= 2x-1 5x+5

29.

f −1 (x)= x+3 2 f −1 (x)= x+3 2

31.

f 1 (x)= x2 f 1 (x)= x2

33.

f 1 (x)= x3 f 1 (x)= x3

35.

f 1 (x)= x3 3 f 1 (x)= x3 3

37.

f 1 (x)= x+4 -2 f 1 (x)= x+4 -2

39.
41.

[-1,0)[1,) [-1,0)[1,)

43.

[-3,0](4,) [-3,0](4,)

45.

[-,-4][-3,3] [-,-4][-3,3]

47.

(2, 0), (0, 1), (8, 2) (2, 0), (0, 1), (8, 2)

49.

(13, 1), (4, 0), (5, 1) (13, 1), (4, 0), (5, 1)

51.

f 1 (x)= xb a 3 f 1 (x)= xb a 3

53.

f 1 (x)= x 2 -b a f 1 (x)= x 2 -b a

55.

f 1 (x)= c x - b a - x f 1 (x)= c x - b a - x

57.

t(h)= 600-h 16 t(h)= 600-h 16 , 3.54 seconds

59.

r(A)= A 4π , ≈ r(A)= A 4π , ≈ 8.92 in.

61.

l(T)=32.2(T2π), ≈ l(T)=32.2(T2π), ≈ 3.26 ft

63.

r(V)= A+ 2π 8π , r(V)= A+ 2π 8π , -2, 3.99 ft

65.

r(V)= V 10π , r(V)= V 10π , ≈ 5.64 ft

5.8 Section Exercises

1.

The graph will have the appearance of a power function.

3.

No. Multiple variables may jointly vary.

5.

y=5 x 2 y=5 x 2

7.

y=10 x 3 y=10 x 3

9.

y=6 x 4 y=6 x 4

11.

y= 18 x 2 y= 18 x 2

13.

y= 81 x 4 y= 81 x 4

15.

y= 20 x 3 y= 20 x 3

17.

y=10xzw y=10xzw

19.

y=10x z y=10x z

21.

y=4 xz w y=4 xz w

23.

y=40 xz w t 2 y=40 xz w t 2

25.

y=256 y=256

27.

y=6 y=6

29.

y=6 y=6

31.

y=27 y=27

33.

y=3 y=3

35.

y=18 y=18

37.

y=90 y=90

39.

y= 81 2 y= 81 2

41.

y= 3 4 x 2 y= 3 4 x 2

Graph of y=3/4(x^2).
43.

y= 1 3 x y= 1 3 x

Graph of y=1/3sqrt(x).
45.

y= 4 x 2 y= 4 x 2

Graph of y=4/(x^2).
47.

1.89 years

49.

0.61 years

51.

3 seconds

53.

48 inches

55.

49.75 pounds

57.

33.33 amperes

59.

2.88 inches

Review Exercises

1.

f(x)= (x2) 2 9vertex (2,–9), intercepts (5,0); (–1,0); (0,–5) f(x)= (x2) 2 9vertex (2,–9), intercepts (5,0); (–1,0); (0,–5)

Graph of f(x)=x^2-4x-5.
3.

f(x)= 3 25 ( x+2 ) 2 +3 f(x)= 3 25 ( x+2 ) 2 +3

5.

300 meters by 150 meters, the longer side parallel to river.

7.

Yes, degree = 5, leading coefficient = 4

9.

Yes, degree = 4, leading coefficient = 1

11.

Asx,f(x),asx,f(x) Asx,f(x),asx,f(x)

13.

–3 with multiplicity 2, 1 2 1 2 with multiplicity 1, –1 with multiplicity 3

15.

4 with multiplicity 1

17.

1 2 1 2 with multiplicity 1, 3 with multiplicity 3

19.

x 2 +4 x 2 +4 with remainder 12

21.

x 2 5x+20 61 x+3 x 2 5x+20 61 x+3

23.

2 x 2 2x3 2 x 2 2x3, so factored form is (x+4)(2 x 2 2x3) (x+4)(2 x 2 2x3)

25.

{ 2, 4,  1 2 } { 2, 4,  1 2 }

27.

{ 1, 3, 4,  1 2 } { 1, 3, 4,  1 2 }

29.

0 or 2 positive, 1 negative

31.

Intercepts (–2,0)and( 0, 2 5 ) (–2,0)and( 0, 2 5 ), Asymptotes x=5 x=5 and y=1. y=1.

Graph of f(x)=(x+1)/(x-5).
33.

Intercepts (3, 0), (-3, 0), and ( 0, 27 2 ) ( 0, 27 2 ), Asymptotes x=1, x=2, y=3. x=1, x=2, y=3.

Graph of f(x)=(3x^2-27)/(x^2+x-2).
35.

y= x2 y= x2

37.

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

39.

f 1 (x)= x+11 3 f 1 (x)= x+11 3

41.

f 1 (x)= (x+3) 2 5 4 ,x3 f 1 (x)= (x+3) 2 5 4 ,x3

43.

y=64 y=64

45.

y = 72 y = 72

47.

148.5 pounds

Practice Test

1.

Degree: 5, leading coefficient: −2

3.

As x−∞, f(x), As x, f(x) As x−∞, f(x), As x, f(x)

5.

f(x)= 3(x-2)2 f(x)=3(x-2)2

7.

3 with multiplicity 3, 1313 with multiplicity 1, 1 with multiplicity 2

9.

-12-12 with multiplicity 3, 2 with multiplicity 2

11.

x3 + 2x2 + 7x + 14 + 26 x-2 x3+2x2+7x+14+ 26 x-2

13.

{-3,-1,32} {-3,-1,32}

15.

1, −2, and −3232 (multiplicity 2)

17.

f(x)= -23(x-3)2 (x-1) (x+2) f(x)=-23(x-3)2(x-1)(x+2)

19.

2 or 0 positive, 1 negative

21.

( -3, 0 ) ( 1, 0 ) ( 0, 3 4 ) (-3,0)(1,0)( 0, 3 4 )

23.

f 1 (x)= (x-4)2 +2,x4 f 1 (x)= (x-4)2+2,x4

25.

f 1 (x)= x+3 3x-2 f 1 (x)= x+3 3x-2

27.

y=20 y=20

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 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/algebra-and-trigonometry/pages/1-introduction-to-prerequisites
  • 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/algebra-and-trigonometry/pages/1-introduction-to-prerequisites
Citation information

© Feb 7, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 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.