Skip to Content
OpenStax Logo
Precalculus

Chapter 2

PrecalculusChapter 2
  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

2.1 Linear Functions

1.

m= 43 02 = 1 2 = 1 2 m= 43 02 = 1 2 = 1 2 ; decreasing because m<0. m<0.

2.

m= 1,8681,442 2,0122,009 = 426 3 =142 people per year m= 1,8681,442 2,0122,009 = 426 3 =142 people per year

3.

y2=2( x+2 ) y2=2( x+2 ) ; y=2x2 y=2x2

4.

y0=3( x0 ) y0=3( x0 ) ; y=3x y=3x

5.

y=7x+3 y=7x+3

6.

H( x )=0.5x+12.5 H( x )=0.5x+12.5

2.2 Graphs of Linear Functions

1.
2.

Possible answers include (3,7),(3,7), (6,9),(6,9), or (9,11).(9,11).

3.
4.

(16, 0)(16, 0)

5.

f(x)=2xf(x)=2x; g(x)= 1 2 x g(x)= 1 2 x

6.

y=13x+6y=13x+6

7.
  1. (0,5)(0,5)
  2. (5, 0)(5, 0)
  3. Slope -1
  4. Neither parallel nor perpendicular
  5. Decreasing function
  6. Given the identity function, perform a vertical flip (over the t-axis) and shift up 5 units.

2.3 Modeling with Linear Functions

1.

C( x )=0.25x+25,000 C( x )=0.25x+25,000 The y-intercept is ( 0,25,000 ). ( 0,25,000 ). If the company does not produce a single doughnut, they still incur a cost of $25,000.

2.

41,100 2020

3.

21.57 miles

2.4 Fitting Linear Models to Data

1.

54°F54°F

2.

150.871 billion gallons; extrapolation

2.1 Section Exercises

1.

Terry starts at an elevation of 3000 feet and descends 70 feet per second.

3.

3 miles per hour

5.

d( t )=10010t d( t )=10010t

7.

Yes.

9.

No.

11.

No.

13.

No.

15.

Increasing.

17.

Decreasing.

19.

Decreasing.

21.

Increasing.

23.

Decreasing.

25.

3

27.

1313

29.

4545

31.

f(x)=12x+72f(x)=12x+72

33.

y=2x+3y=2x+3

35.

y=13x+223y=13x+223

37.

y=45x+4y=45x+4

39.

5454

41.

y= 2 3 x+1 y= 2 3 x+1

43.

y=2x+3 y=2x+3

45.

y=3 y=3

47.

Linear, g(x)=3x+5g(x)=3x+5

49.

Linear, f(x)=5x5f(x)=5x5

51.

Linear, g(x)=252x+6g(x)=252x+6

53.

Linear, f(x)=10x24f(x)=10x24

55.

f(x)=58x+17.3 f(x)=58x+17.3

57.
59.

a. a=11,900a=11,900; b=1001.1b=1001.1 b. q(p)=1000p100q(p)=1000p100

61.
..
63.

x=163x=163

65.

x=ax=a

67.

y=dcaxadcay=dcaxadca

69.

$45 per training session.

71.

The rate of change is 0.1. For every additional minute talked, the monthly charge increases by $0.1 or 10 cents. The initial value is 24. When there are no minutes talked, initially the charge is $24.

73.

The slope is 400.400. This means for every year between 1960 and 1989, the population dropped by 400 per year in the city.

75.

c.

2.2 Section Exercises

1.

The slopes are equal; y-intercepts are not equal.

3.

The point of intersection is (a,a).(a,a). This is because for the horizontal line, all of the yy coordinates are aa and for the vertical line, all of the xx coordinates are a.a. The point of intersection will have these two characteristics.

5.

First, find the slope of the linear function. Then take the negative reciprocal of the slope; this is the slope of the perpendicular line. Substitute the slope of the perpendicular line and the coordinate of the given point into the equation y=mx+by=mx+b and solve for b.b. Then write the equation of the line in the form y=mx+by=mx+b by substituting in mm and b.b.

7.

neither parallel or perpendicular

9.

perpendicular

11.

parallel

13.

(20)(20); (0, 4)(0, 4)

15.

(150)(150); (0, 1)(0, 1)

17.

(80)(80); (028)(028)

19.

Line 1: m=8   Line 2: m=6   Neither Line 1: m=8   Line 2: m=6   Neither

21.

Line 1: m= 1 2    Line 2: m=2   Perpendicular Line 1: m= 1 2    Line 2: m=2   Perpendicular

23.

Line 1: m=2   Line 2: m=2   Parallel Line 1: m=2   Line 2: m=2   Parallel

25.

g(x)=3x3g(x)=3x3

27.

p(t)=13t+2p(t)=13t+2

29.

(2,1)(2,1)

31.

(175,53)(175,53)

33.

F

35.

C

37.

A

39.
41.
43.
45.
47.
49.
51.
53.
55.
57.
59.

g(x)=0.75x5.5 g(x)=0.75x5.5 ; 0.75; (0,5.5)(0,5.5)

61.

y=3y=3

63.

x=3x=3

65.

no point of intersection

67.

(2, 7) (2, 7)

69.

(10, –5)(10, –5)

71.

y=100x98 y=100x98

73.

x< 1999 201 x> 1999 201 x< 1999 201 x> 1999 201

75.

Less than 3000 texts

2.3 Section Exercises

1.

Determine the independent variable. This is the variable upon which the output depends.

3.

To determine the initial value, find the output when the input is equal to zero.

5.

6 square units

7.

20.012 square units

9.

2,300

11.

64,170

13.

P( t )=75,000+2,500t P( t )=75,000+2,500t

15.

(–30, 0) Thirty years before the start of this model, the town had no citizens. (0, 75,000) Initially, the town had a population of 75,000.

17.

Ten years after the model began.

19.

W( t )=0.5t+7.5 W( t )=0.5t+7.5

21.

( 15,0 ) ( 15,0 ) : The x-intercept is not a plausible set of data for this model because it means the baby weighed 0 pounds 15 months prior to birth. ( 0, 7.5 ) ( 0, 7.5 ) : The baby weighed 7.5 pounds at birth.

23.

At age 5.8 months.

25.

C( t )=12,025205t C( t )=12,025205t

27.

(58.7, 0) (58.7, 0) : In roughly 59 years, the number of people inflicted with the common cold would be 0. (0,12,025) (0,12,025) : Initially there were 12,025 people afflicted by the common cold.

29.

2064

31.

y=2t+180 y=2t+180

33.

In 2070, the company’s profit will be zero.

35.

y=30t300 y=30t300

37.

(10, 0) In 1990, the profit earned zero profit.

39.

Hawaii

41.

During the year 1933

43.

$105,620

45.
  1. 696 people
  2. 4 years
  3. 174 people per year
  4. 305 people
  5. P(t)=305+174t P(t)=305+174t
  6. 2,219 people
47.
  1. C( x )=0.15x+10 C( x )=0.15x+10
  2. The flat monthly fee is $10 and there is an additional $0.15 fee for each additional minute used
  3. $113.05
49.
  1. P( t )=190t+4360 P( t )=190t+4360
  2. 6,640 moose
51.
  1. R( t )=162.1t R( t )=162.1t
  2. 5.5 billion cubic feet
  3. During the year 2017
53.

More than 133 minutes

55.

More than $42,857.14 worth of jewelry

57.

$66,666.67

2.4 Section Exercises

1.

When our model no longer applies, after some value in the domain, the model itself doesn’t hold.

3.

We predict a value outside the domain and range of the data.

5.

The closer the number is to 1, the less scattered the data, the closer the number is to 0, the more scattered the data.

7.

61.966 years

9.

No.

11.

No.

13.

Interpolation. About 60° F. 60° F.

15.

C

17.

B

19.
21.
23.

Yes, trend appears linear because r=0.985r=0.985 and will exceed 12,000 near midyear, 2016, 24.6 years since 1992.

25.

y=1.640x+13.800y=1.640x+13.800, r=0.987r=0.987

27.

y=0.962x+26.86,   r=0.965 y=0.962x+26.86,   r=0.965

29.

y=1.981x+60.197y=1.981x+60.197; r=0.998r=0.998

31.

y=0.121x38.841,r=0.998y=0.121x38.841,r=0.998

33.

(−2,−6),(1,−12),(5,−20),(6,−22),(9,−28)(−2,−6),(1,−12),(5,−20),(6,−22),(9,−28); y=−2x−10y=−2x−10

35.

(189.8,0) (189.8,0)  If 18,980 units are sold, the company will have a profit of zero dollars.

37.

y=0.00587x+1985.41y=0.00587x+1985.41

39.

y=20.25x671.5y=20.25x671.5

41.

y=10.75x+742.50y=10.75x+742.50

Review Exercises

1.

Yes

3.

Increasing.

5.

y=3x+26y=3x+26

7.

3

9.

y=2x2y=2x2

11.

Not linear.

13.

parallel

15.

(–9,0);(0,–7)(–9,0);(0,–7)

17.

Line 1: m=2;m=2; Line 2: m=2;m=2; Parallel

19.

y=0.2x+21y=0.2x+21

21.
23.

250.

25.

118,000.

27.

y=300x+11,500y=300x+11,500

29.

a) 800; b) 100 students per year; c) P(t)=100t+1700P(t)=100t+1700

31.

18,500

33.

$91,625

35.

Extrapolation.

37.
39.

Midway through 2024.

41.

y=1.294x+49.412;  r=0.974y=1.294x+49.412;  r=0.974

43.

Early in 2022

45.

7,660

Practice Test

1.

Yes.

3.

Increasing

5.

y=−1.5x6y=−1.5x6

7.

y=2x1y=2x1

9.

No.

11.

Perpendicular

13.

(7,0)(7,0); (0,2)(0,2)

15.

y=0.25x+12y=0.25x+12

17.
19.

150

21.

165,000

23.

y=875x+10,675y=875x+10,675

25.

a) 375; b) dropped an average of 46.875, or about 47 people per year; c) y=46.875t+1250y=46.875t+1250

27.
29.

Early in 2018

31.

y=0.00455x+1979.5y=0.00455x+1979.5

33.

r=0.999r=0.999

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/precalculus/pages/1-introduction-to-functions
  • 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/precalculus/pages/1-introduction-to-functions
Citation information

© Feb 10, 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.