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Precalculus

Chapter 3

PrecalculusChapter 3
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  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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. 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. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. 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. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. 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 Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. 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

Try It

3.1 Complex Numbers

1.

24 =0+2i 6 24 =0+2i 6

2.
Graph of the plotted point, -4-i.
3.

(34i)(2+5i)=19i (34i)(2+5i)=19i

4.

824i 824i

5.

18+i 18+i

6.

10229i 10229i

7.

3 17 + 5i 17 3 17 + 5i 17

3.2 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

3.3 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.

3.4 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).

3.5 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

3.6 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

3.7 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.

3.8 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

3.9 Modeling Using Variation

1.

128 3 128 3

2.

9 2 9 2

3.

x=20 x=20

3.1 Section Exercises

1.

Add the real parts together and the imaginary parts together.

3.

i i times i i equals –1, which is not imaginary. (answers vary)

5.

8+2i 8+2i

7.

14+7i 14+7i

9.

23 29 + 15 29 i 23 29 + 15 29 i

11.

2 real and 0 nonreal

13.
Graph of the plotted point, 1-2i.
15.
Graph of the plotted point, i.
17.

8i 8i

19.

11+4i 11+4i

21.

25i 25i

23.

6+15i 6+15i

25.

16+32i 16+32i

27.

47i 47i

29.

25

31.

2 2 3 i 2 2 3 i

33.

46i 46i

35.

2 5 + 11 5 i 2 5 + 11 5 i

37.

15i 15i

39.

1+i 3 1+i 3

41.

1 1

43.

1 1

45.

128i

47.

( 3 2 + 1 2 i ) 6 =1 ( 3 2 + 1 2 i ) 6 =1

49.

3i 3i

51.

0

53.

5 – 5i

55.

2i 2i

57.

9 2 9 2 i 9 2 9 2 i

3.2 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.

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

{ 2i 2 ,2i 2 } { 2i 2 ,2i 2 }

29.

{ 3i 3 ,3i 3 } { 3i 3 ,3i 3 }

31.

{2+i,2i} {2+i,2i}

33.

{2+3i,23i} {2+3i,23i}

35.

{5+i,5i} {5+i,5i}

37.

{2+2 6 , 22 6 } {2+2 6 , 22 6 }

39.

{ 1 2 + 3 2 i,  1 2 3 2 i } { 1 2 + 3 2 i,  1 2 3 2 i }

41.

{ 3 5 + 1 5 i,  3 5 1 5 i } { 3 5 + 1 5 i,  3 5 1 5 i }

43.

{ 1 2 + 1 2 i 7 ,  1 2 1 2 i 7 } { 1 2 + 1 2 i 7 ,  1 2 1 2 i 7 }

45.

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

47.

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

49.

f(x)= 6 49 x 2 + 60 49 x+ 297 49 f(x)= 6 49 x 2 + 60 49 x+ 297 49

51.

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

53.
Graph of f(x) = x^2-2x

Vertex ( 1, 1 ), ( 1, 1 ), Axis of symmetry is x=1. x=1. Intercepts are (0,0), (2,0). (0,0), (2,0).

55.
Graph of f(x)x^2-5x-6

Vertex ( 5 2 , 49 4 ), ( 5 2 , 49 4 ), Axis of symmetry is x= 5 2 , x= 5 2 , intercepts: (6,0), (1,0). (6,0), (1,0).

57.
Graph of f(x)=-2x^2+5x-8

Vertex ( 5 4 ,  39 8 ), ( 5 4 ,  39 8 ), Axis of symmetry is x= 5 4 . x= 5 4 . Intercepts are ( 0, 8 ). ( 0, 8 ).

59.

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

61.

f(x)=2 x 2 +8x1 f(x)=2 x 2 +8x1

63.

f(x)= 1 2 x 2 3x+ 7 2 f(x)= 1 2 x 2 3x+ 7 2

65.

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

67.

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

69.

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

71.

The graph is shifted up or down (a vertical shift).

73.

50 feet

75.

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

77.

Domain is (,) (,) Range is (,11]. (,11].

79.

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

81.

f(x)=3 x 2 9 f(x)=3 x 2 9

83.

f(x)=5 x 2 77 f(x)=5 x 2 77

85.

50 feet by 50 feet. Maximize f(x)= x 2 +100x. f(x)= x 2 +100x.

87.

125 feet by 62.5 feet. Maximize f(x)=2 x 2 +250x. f(x)=2 x 2 +250x.

89.

6 6 and 6; 6; product is –36; maximize f(x)= x 2 +12x. f(x)= x 2 +12x.

91.

2909.56 meters

93.

$10.70

3.3 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.

f(x)f(x) is a power function because it contains a variable base raised to a fixed power. It is also a polynomial, with all coefficients except one equal to zero.

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.

44.

No.

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

3.4 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 2 3 with multiplicity 5,55,5 with multiplicity 2 2

37.

0 0 with multiplicity 4,24,2 with multiplicity 1,11,1 with multiplicity 1 1

39.

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

41.

0 0 with multiplicity 6, 2 3 6, 2 3 with multiplicity 2 2

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 xx, g(x)g(x), as xx, g(x)g(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 xx, k(x)k(x), as x x, k(x). k(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 x, n(x)n(x), as xx, n(x). n(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)

3.5 Section Exercises

1.

The binomial is a factor of the polynomial.

3.

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

5.

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

7.

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

9.

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

11.

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

13.

2 x 2 3x+5 2 x 2 3x+5 , quotient: 2 x 2 3x+5 2 x 2 3x+5, remainder: 0 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+164 x 2 +8x+16, remainder: 1 1

51.

Quotient: 3 x 2 +3x+53 x 2 +3x+5, remainder: 0 0

53.

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

3.6 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

3.7 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),x 5 2 ,f(x),x 5 2 + , f(x) f(x)


End behavior: x±, x±, f(x) 1 3 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=9 x2 x 2 9 y=9 x2 x 2 9

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.

3.8 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)= x5 3 f 1 (x)= x5 3

11.

f(x)= 9x f(x)= 9x

13.

f 1 (x)= x5 3 f 1 (x)= x5 3

15.

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

17.

f 1 (x)= x 2 1 2 ,[ 0, ) f 1 (x)= x 2 1 2 ,[ 0, )

19.

f 1 (x)= ( x9 ) 2 +4 4 ,[ 9, ) f 1 (x)= ( x9 ) 2 +4 4 ,[ 9, )

21.

f 1 (x)= ( x9 2 ) 3 f 1 (x)= ( x9 2 ) 3

23.

f 1 (x)= 28x x f 1 (x)= 28x x

25.

f 1 (x)= 7x3 1x f 1 (x)= 7x3 1x

27.

f 1 (x)= 5x4 4x+3 f 1 (x)= 5x4 4x+3

29.

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

31.

f 1 (x)= x+6 +3 f 1 (x)= x+6 +3

33.

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

Graph of f(x)=4- x^2 and its inverse, f^(-1)(x)= sqrt(4-x).
35.

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

Graph of f(x)= (x-4)^2 and its inverse, f^(-1)(x)= sqrt(x)+4.
37.

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

Graph of f(x)= 1-x^3 and its inverse, f^(-1)(x)= (1-x)^(1/3).
39.

f 1 (x)= x+8 +3 f 1 (x)= x+8 +3

Graph of f(x)= x^2-6x+1 and its inverse, f^(-1)(x)= sqrt(x+8)+3.
41.

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

Graph of f(x)= 1/x^2 and its inverse, f^(-1)(x)= sqrt(1/x).
43.

[2,1)[3,) [2,1)[3,)

Graph of f(x)= sqrt((x+2)(x-3)/(x-1)).
45.

[4,2)[5,) [4,2)[5,)

Graph of f(x)= sqrt((x^2-x-20)/(x-2)).
47.

(2, 0); (4, 2); (22, 3) (2, 0); (4, 2); (22, 3)

Graph of f(x)= x^3-x-2.
49.

(4, 0); (0, 1); (10, 2) (4, 0); (0, 1); (10, 2)

Graph of f(x)= x^3+3x-4.
51.

(3, 1); (1, 0); (7, 1) (3, 1); (1, 0); (7, 1)

Graph of f(x)= x^4+5x+1.
53.

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

55.

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

57.

t(h)= 200h 4.9 , t(h)= 200h 4.9 , 5.53 seconds

59.

r(V)= 3V 4π 3 , r(V)= 3V 4π 3 , 3.63 feet

61.

n(C)= 100C25 0.6C , n(C)= 100C25 0.6C , 250 mL

63.

r(V)= V 6π , r(V)= V 6π , 3.99 meters

65.

r(V)= V 4π , r(V)= V 4π , 1.99 inches

3.9 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=11944 x 3 y=11944 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.

22i 22i

3.

24+3i 24+3i

5.

{2+i, 2i} {2+i, 2i}

7.

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.
9.

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

11.

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

13.

Yes, degree = 5, leading coefficient = 4

15.

Yes, degree = 4, leading coefficient = 1

17.

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

19.

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

21.

4 with multiplicity 1

23.

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

25.

x 2 +4 x 2 +4 with remainder 12

27.

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

29.

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

31.

{ 2, 4,  1 2 } { 2, 4,  1 2 }

33.

{ 1, 3, 4,  1 2 } { 1, 3, 4,  1 2 }

35.

0 or 2 positive, 1 negative

37.

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).
39.

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).
41.

y= x2 y= x2

43.

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

45.

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

47.

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

49.

y=64 y=64

51.

y = 72 y = 72

53.

148.5 pounds

Practice Test

1.

2010i 2010i

3.

{2+3i, 23i} {2+3i, 23i}

5.

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

7.

f(x)= ( x+1 ) 2 9 f(x)= ( x+1 ) 2 9, vertex ( −1,−9 ) ( −1,−9 ), intercepts ( 2,0 );( −4,0 );( 0,−8 ) ( 2,0 );( −4,0 );( 0,−8 )

Graph of f(x)=x^2+2x-8.
9.

60,000 square feet

11.

0 with multiplicity 4, 3 with multiplicity 2

13.

2 x 2 4x+11 26 x+2 2 x 2 4x+11 26 x+2

15.

2 x 2 x4 2 x 2 x4. So factored form is (x+3)(2 x 2 x4) (x+3)(2 x 2 x4)

17.

1 2 1 2 (has multiplicity 2), 1±i 15 2 1±i 15 2

19.

2 2(has multiplicity 3), ±i ±i

21.

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

23.

Intercepts (4,0),( 0, 4 3 ) (4,0),( 0, 4 3 ) , Asymptotes x=3, x =−1, y=0 x=3, x =−1, y=0.

Graph of f(x)=(x+4)/(x^2-2x-3).
25.

y=x+4 y=x+4

27.

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

29.

y=18 y=18

31.

4 seconds

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