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Precalculus

Chapter 1

PrecalculusChapter 1
  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

1.1 Functions and Function Notation

1.

a. yes; b. yes. (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

2.

w=f(d) w=f(d)

3.

yes

4.

g( 5 )=1 g( 5 )=1

5.

m=8 m=8

6.

y=f( x )= x 3 2 y=f( x )= x 3 2

7.

g( 1 )=8 g( 1 )=8

8.

x=0 x=0or x=2 x=2

9.

a. yes, because each bank account has a single balance at any given time; b. no, because several bank account numbers may have the same balance; c. no, because the same output may correspond to more than one input.

10.
  1. Yes, letter grade is a function of percent grade;
  2. No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.
11.

yes

12.

No, because it does not pass the horizontal line test.

1.2 Domain and Range

1.

{5,0,5,10,15} {5,0,5,10,15}

2.

( , ) ( , )

3.

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

4.

[ 5 2 , ) [ 5 2 , )

5.
  1. values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3;
  2. { x|x2or1x<3 } { x|x2or1x<3 } ;
  3. (,2][1,3) (,2][1,3)
6.

domain =[1950,2002] range = [47,000,000,89,000,000]

7.

domain: ( ,2 ]; ( ,2 ];range: ( ,0 ] ( ,0 ]

8.
Graph of f(x).

1.3 Rates of Change and Behavior of Graphs

1.

$2.84$2.31 5 years = $0.53 5 years =$0.106 $2.84$2.31 5 years = $0.53 5 years =$0.106per year.

2.

1 2 1 2

3.

a+7 a+7

4.

The local maximum appears to occur at (1,28), (1,28),and the local minimum occurs at (5,80). (5,80).The function is increasing on (,1)(5,) (,1)(5,)and decreasing on (1,5). (1,5).

Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).

1.4 Composition of Functions

1.

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

No, the functions are not the same.

2.

A gravitational force is still a force, so a( G(r) ) a( G(r) ) makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G( a(F) ) G( a(F) ) does not make sense.

3.

f(g(1))=f(3)=3 f(g(1))=f(3)=3and g(f(4))=g(1)=3 g(f(4))=g(1)=3

4.

g(f(2))=g(5)=3 g(f(2))=g(5)=3

5.

a. 8; b. 20

6.

[ 4,0 )( 0, ) [ 4,0 )( 0, )

7.

Possible answer:

g( x )= 4+ x 2 g( x )= 4+ x 2
h( x )= 4 3x h( x )= 4 3x
f=hg f=hg

1.5 Transformation of Functions

1.
b(t)=h(t)+10=4.9 t 2 +30t+10 b(t)=h(t)+10=4.9 t 2 +30t+10
2.

The graphs of f(x) f(x)and g(x) g(x)are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.

Graph of a square root function and a horizontally shift square foot function.
3.
Graph of h(x)=|x-2|+4.
4.

g( x )= 1 x-1 +1 g( x )= 1 x-1 +1

5.
  1. Graph of a vertically reflected absolute function.
  2. Graph of an absolute function translated one unit left.
6.
  1. g(x)=f(x) g(x)=f(x)

    x x -2 0 2 4
    g(x) g(x) 5 5 10 10 15 15 20 20
  2. h(x)=f(x) h(x)=f(x)

    x x -2 0 2 4
    h(x) h(x) 15 10 5 unknown
7.
Graph of x^2 and its reflections.

Notice: g(x)=f(x) g(x)=f(x)looks the same as f(x) f(x).

8.

even

9.
x x 2 4 6 8
g(x) g(x) 9 12 15 0
10.

g(x)=3x-2 g(x)=3x-2

11.

g(x)=f( 1 3 x ) g(x)=f( 1 3 x )so using the square root function we get g(x)= 1 3 x g(x)= 1 3 x

1.6 Absolute Value Functions

1.

| x2 |3 | x2 |3

2.

using the variable p pfor passing, | p80 |20 | p80 |20

3.

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

4.

x=1 x=1or x=2 x=2

5.

f(0)=1, f(0)=1,so the graph intersects the vertical axis at (0,1). (0,1). f(x)=0 f(x)=0when x=5 x=5and x=1 x=1so the graph intersects the horizontal axis at (5,0) (5,0)and (1,0). (1,0).

6.

-8x4 -8x4

7.

k1 k1or k7; k7;in interval notation, this would be (,1][7,) (,1][7,)

1.7 Inverse Functions

1.

h(2)=6 h(2)=6

2.

No

3.

Yes

4.

The domain of function f 1 f 1 is (,2) (,2) and the range of function f 1 f 1 is (1,). (1,).

5.
  1. f(60)=50. f(60)=50.In 60 minutes, 50 miles are traveled.
  2. f 1 (60)=70. f 1 (60)=70.To travel 60 miles, it will take 70 minutes.
6.

a. 3; b. 5.6

7.

x=3y+5 x=3y+5

8.

f 1 (x)= ( 2x ) 2 ;domainoff:[ 0, );domainof f 1 :( ,2 ] f 1 (x)= ( 2x ) 2 ;domainoff:[ 0, );domainof f 1 :( ,2 ]

9.
Graph of f(x) and f^(-1)(x).

1.1 Section Exercises

1.

A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.

3.

When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.

5.

When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.

7.

function

9.

function

11.

function

13.

function

15.

function

17.

function

19.

function

21.

function

23.

function

25.

not a function

27.

f(3)=11; f(2)=1; f(a)=2a5; f(a)=2a+5;f(a+h)=2a+2h5 f(3)=11; f(2)=1; f(a)=2a5; f(a)=2a+5;f(a+h)=2a+2h5

29.

f(3)= 5 +5; f(2)=5; f(a)= 2+a +5; f(a)= 2a 5;f(a+h)= f(3)= 5 +5; f(2)=5; f(a)= 2+a +5; f(a)= 2a 5;f(a+h)= 2ah +5 2ah +5

31.

f(3)=2; f(2)=13=2; f(a)=| a1 || a+1 |; f(a)=| a1 |+| a+1 |; f(a+h)=| a+h1 || a+h+1 | f(3)=2; f(2)=13=2; f(a)=| a1 || a+1 |; f(a)=| a1 |+| a+1 |; f(a+h)=| a+h1 || a+h+1 |

33.

g(x)g(a) xa =x+a+2,xa g(x)g(a) xa =x+a+2,xa

35.

a. f(2)=14; f(2)=14;b. x=3 x=3

37.

a. f(5)=10; f(5)=10;b. x=1  x=1 or  x=4  x=4

39.

a. f(t)=6 2 3 t; f(t)=6 2 3 t;b. f(3)=8; f(3)=8;c. t=6 t=6

41.

not a function

43.

function

45.

function

47.

function

49.

function

51.

function

53.

a. f(0)=1; f(0)=1;b. f(x)=3,x=2  f(x)=3,x=2 or  x=2  x=2

55.

not a function so it is also not a one-to-one function

57.

one-to- one function

59.

function, but not one-to-one

61.

function

63.

function

65.

not a function

67.

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

69.

f(2)=14; f(1)=11; f(0)=8; f(1)=5; f(2)=2 f(2)=14; f(1)=11; f(0)=8; f(1)=5; f(2)=2

71.

f(2)=4;   f(1)=4.414; f(0)=4.732; f(1)=4.5; f(2)=5.236 f(2)=4;   f(1)=4.414; f(0)=4.732; f(1)=4.5; f(2)=5.236

73.

f(2)= 1 9 ; f(1)= 1 3 ; f(0)=1; f(1)=3; f(2)=9 f(2)= 1 9 ; f(1)= 1 3 ; f(0)=1; f(1)=3; f(2)=9

75.

20

77.

[0, 100] [0, 100]

Graph of a parabola.
79.

[0.001, 0.001] [0.001, 0.001]

Graph of a parabola.
81.

[1,000,000, 1,000,000] [1,000,000, 1,000,000]

Graph of a cubic function.
83.

[0, 10] [0, 10]

Graph of a square root function.
85.

[−0.1,0.1] [−0.1,0.1]

Graph of a square root function.
87.

[100, 100] [100, 100]

Graph of a cubic root function.
89.

a. g(5000)=50; g(5000)=50; b. The number of cubic yards of dirt required for a garden of 100 square feet is 1.

91.

a. The height of a rocket above ground after 1 second is 200 ft. b. the height of a rocket above ground after 2 seconds is 350 ft.

1.2 Section Exercises

1.

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

3.

There is no restriction on x xfor f(x)= x 3 f(x)= x 3 because you can take the cube root of any real number. So the domain is all real numbers, (,). (,).When dealing with the set of real numbers, you cannot take the square root of negative numbers. So x x-values are restricted for f(x)= x f(x)= x to nonnegative numbers and the domain is [0,). [0,).

5.

Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the x x-axis and y y-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate or  .  . Combine the graphs to find the graph of the piecewise function.

7.

(,) (,)

9.

(,3] (,3]

11.

(,) (,)

13.

(,) (,)

15.

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

17.

(,11)(11,2)(2,) (,11)(11,2)(2,)

19.

(,3)(3,5)(5,) (,3)(3,5)(5,)

21.

(,5) (,5)

23.

[6,) [6,)

25.

( ,9 )( 9,9 )( 9, ) ( ,9 )( 9,9 )( 9, )

27.

domain: (2,8], (2,8],range [6,8) [6,8)

29.

domain: [4, 4], [4, 4],range: [0, 2] [0, 2]

31.

domain: [5, 3), [5, 3),range: [ 0,2 ] [ 0,2 ]

33.

domain: (,1], (,1],range: [0,) [0,)

35.

domain: [ 6, 1 6 ][ 1 6 ,6 ]; [ 6, 1 6 ][ 1 6 ,6 ];range: [ 6, 1 6 ][ 1 6 ,6 ] [ 6, 1 6 ][ 1 6 ,6 ]

37.

domain: [3, ); [3, );range: [0,) [0,)

39.

domain: (,) (,)

Graph of f(x).
41.

domain: (,) (,)

Graph of f(x).
43.

domain: (,) (,)

Graph of f(x).
45.

domain: (,) (,)

Graph of f(x).
47.

f(3)=1; f(2)=0; f(1)=0; f(0)=0 f(3)=1; f(2)=0; f(1)=0; f(0)=0

49.

f(1)=4; f(0)=6; f(2)=20; f(4)=34 f(1)=4; f(0)=6; f(2)=20; f(4)=34

51.

f(1)=5; f(0)=3; f(2)=3; f(4)=16 f(1)=5; f(0)=3; f(2)=3; f(4)=16

53.

domain: (,1)(1,) (,1)(1,)

55.
Graph of the equation from [-0.5, -0.1].

window: [0.5,0.1]; [0.5,0.1];range: [4, 100] [4, 100]

Graph of the equation from [0.1, 0.5].

window: [0.1, 0.5]; [0.1, 0.5];range: [4, 100] [4, 100]

57.

[0, 8] [0, 8]

59.

Many answers. One function is f(x)= 1 x2 . f(x)= 1 x2 .

1.3 Section Exercises

1.

Yes, the average rate of change of all linear functions is constant.

3.

The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

5.

4( b+1 ) 4( b+1 )

7.

3

9.

4x+2h 4x+2h

11.

1 13( 13+h ) 1 13( 13+h )

13.

3 h 2 +9h+9 3 h 2 +9h+9

15.

4x+2h3 4x+2h3

17.

4 3 4 3

19.

increasing on ( ,2.5 )( 1, ), ( ,2.5 )( 1, ),decreasing on (2.5, 1) (2.5, 1)

21.

increasing on ( ,1 )( 3,4 ), ( ,1 )( 3,4 ),decreasing on ( 1,3 )( 4, ) ( 1,3 )( 4, )

23.

local maximum: (3, 50), (3, 50),local minimum: (3, 50) (3, 50)

25.

absolute maximum at approximately (7, 150), (7, 150),absolute minimum at approximately (−7.5, −220) (−7.5, −220)

27.

a. –3000; b. –1250

29.

-4

31.

27

33.

–0.167

35.

Local minimum at (3,22), (3,22),decreasing on (, 3), (, 3),increasing on (3, ) (3, )

37.

Local minimum at (2,2), (2,2),decreasing on (3,2), (3,2),increasing on (2, ) (2, )

39.

Local maximum at (0.5, 6), (0.5, 6),local minima at (3.25,47) (3.25,47)and (2.1,32), (2.1,32),decreasing on (,3.25) (,3.25)and (0.5, 2.1), (0.5, 2.1),increasing on (3.25, 0.5) (3.25, 0.5)and (2.1, ) (2.1, )

41.

A

43.

b=5 b=5

45.

2.7 gallons per minute

47.

approximately –0.6 milligrams per day

1.4 Section Exercises

1.

Find the numbers that make the function in the denominator g gequal to zero, and check for any other domain restrictions on f fand g, g,such as an even-indexed root or zeros in the denominator.

3.

Yes. Sample answer: Let f(x)=x+1 and g(x)=x1. f(x)=x+1 and g(x)=x1.Then f(g(x))=f(x1)=(x1)+1=x f(g(x))=f(x1)=(x1)+1=xand g(f(x))=g(x+1)=(x+1)1=x. g(f(x))=g(x+1)=(x+1)1=x.So fg=gf. fg=gf.

5.

(f+g)( x )=2x+6, (f+g)( x )=2x+6,domain: (,) (,)

(fg)( x )=2 x 2 +2x6, (fg)( x )=2 x 2 +2x6,domain: (,) (,)

(fg)( x )= x 4 2 x 3 +6 x 2 +12x, (fg)( x )= x 4 2 x 3 +6 x 2 +12x,domain: (,) (,)

( f g )( x )= x 2 +2x 6 x 2 , ( f g )( x )= x 2 +2x 6 x 2 ,domain: (, 6 )( 6 , 6 )( 6 ,) (, 6 )( 6 , 6 )( 6 ,)

7.

(f+g)( x )= 4 x 3 +8 x 2 +1 2x , (f+g)( x )= 4 x 3 +8 x 2 +1 2x ,domain: (,0)(0,) (,0)(0,)

(fg)( x )= 4 x 3 +8 x 2 1 2x , (fg)( x )= 4 x 3 +8 x 2 1 2x ,domain: (,0)(0,) (,0)(0,)

(fg)( x )=x+2, (fg)( x )=x+2,domain: (,0)(0,) (,0)(0,)

( f g )( x )=4 x 3 +8 x 2 , ( f g )( x )=4 x 3 +8 x 2 ,domain: (,0)(0,) (,0)(0,)

9.

(f+g)(x)=3 x 2 + x5 , (f+g)(x)=3 x 2 + x5 ,domain: [5,) [5,)

(fg)(x)=3 x 2 x5 , (fg)(x)=3 x 2 x5 ,domain: [5,) [5,)

(fg)(x)=3 x 2 x5 , (fg)(x)=3 x 2 x5 ,domain: [5,) [5,)

( f g )(x)= 3 x 2 x5 , ( f g )(x)= 3 x 2 x5 ,domain: (5,) (5,)

11.

a. 3; b. f( g( x ) )=2 ( 3x5 ) 2 +1; f( g( x ) )=2 ( 3x5 ) 2 +1;c. g( f)( x ) )=6 x 2 2; g( f)( x ) )=6 x 2 2;d. ( gg )(x)=3(3x5)5=9x20; ( gg )(x)=3(3x5)5=9x20;e. ( ff )( 2 )=163 ( ff )( 2 )=163

13.

f(g(x))= x 2 +3 +2,g(f(x))=x+4 x +7 f(g(x))= x 2 +3 +2,g(f(x))=x+4 x +7

15.

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

17.

( fg )(x)= 1 2 x +44 = x 2 , ( gf )(x)=2x4 ( fg )(x)= 1 2 x +44 = x 2 , ( gf )(x)=2x4

19.

f(g(h(x)))= ( 1 x+3 ) 2 +1 f(g(h(x)))= ( 1 x+3 ) 2 +1

21.

a. (gf)(x)= 3 24x ; (gf)(x)= 3 24x ;b. ( , 1 2 ) ( , 1 2 )

23.

a. (0,2)(2,); (0,2)(2,);b. (,2)(2,); (,2)(2,);c. (0,) (0,)

25.

(1,) (1,)

27.

sample: f(x)= x 3 g(x)=x5 f(x)= x 3 g(x)=x5

29.

sample: f(x)= 4 x g(x)= (x+2) 2 f(x)= 4 x g(x)= (x+2) 2

31.

sample: f(x)= x 3 g(x)= 1 2x3 f(x)= x 3 g(x)= 1 2x3

33.

sample: f(x)= x 4 g(x)= 3x2 x+5 f(x)= x 4 g(x)= 3x2 x+5

35.

sample: f(x)= x f(x)= x
g(x)=2x+6 g(x)=2x+6

37.

sample: f(x)= x 3 f(x)= x 3
g(x)=(x1) g(x)=(x1)

39.

sample: f(x)= x 3 f(x)= x 3
g(x)= 1 x2 g(x)= 1 x2

41.

sample: f(x)= x f(x)= x
g(x)= 2x1 3x+4 g(x)= 2x1 3x+4

43.

2

45.

5

47.

4

49.

0

51.

2

53.

1

55.

4

57.

4

59.

9

61.

4

63.

2

65.

3

67.

11

69.

0

71.

7

73.

f(g(0))=27,g( f(0) )=94 f(g(0))=27,g( f(0) )=94

75.

f(g(0))= 1 5 ,g(f(0))=5 f(g(0))= 1 5 ,g(f(0))=5

77.

18 x 2 +60x+51 18 x 2 +60x+51

79.

gg(x)=9x+20 gg(x)=9x+20

81.

2

83.

(,) (,)

85.

False

87.

(fg)(6)=6 (fg)(6)=6; (gf)(6)=6 (gf)(6)=6

89.

(fg)(11)=11,(gf)(11)=11 (fg)(11)=11,(gf)(11)=11

91.

c

93.

A(t)=π ( 25 t+2 ) 2 A(t)=π ( 25 t+2 ) 2 and A(2)=π ( 25 4 ) 2 =2500π A(2)=π ( 25 4 ) 2 =2500π square inches

95.

A(5)=π ( 2(5)+1 ) 2 =121π A(5)=π ( 2(5)+1 ) 2 =121πsquare units

97.

a. N(T(t))=23 (5t+1.5) 2 56(5t+1.5)+1; N(T(t))=23 (5t+1.5) 2 56(5t+1.5)+1; b. 3.38 hours

1.5 Section Exercises

1.

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

3.

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

5.

For a function f, f,substitute (x) (x)for (x) (x)in f(x). f(x).Simplify. If the resulting function is the same as the original function, f(x)=f(x), f(x)=f(x),then the function is even. If the resulting function is the opposite of the original function, f(x)=f(x), f(x)=f(x),then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

7.

g(x)=|x-1|3 g(x)=|x-1|3

9.

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

11.

The graph of f(x+43) f(x+43)is a horizontal shift to the left 43 units of the graph of f. f.

13.

The graph of f(x-4) f(x-4)is a horizontal shift to the right 4 units of the graph of f. f.

15.

The graph of f(x)+8 f(x)+8is a vertical shift up 8 units of the graph of f. f.

17.

The graph of f(x)7 f(x)7is a vertical shift down 7 units of the graph of f. f.

19.

The graph of f(x+4)1 f(x+4)1 is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of f. f.

21.

decreasing on (,3) (,3)and increasing on (3,) (3,)

23.

decreasing on (0,) (0,)

25.
Graph of k(x).
27.
Graph of f(t).
29.
Graph of k(x).
31.

g(x)=f(x-1),h(x)=f(x)+1 g(x)=f(x-1),h(x)=f(x)+1

33.

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

35.

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

37.

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

39.

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

41.

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

43.

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

45.

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

47.

even

49.

odd

51.

even

53.

The graph of g gis a vertical reflection (across the x x-axis) of the graph of f. f.

55.

The graph of g gis a vertical stretch by a factor of 4 of the graph of f. f.

57.

The graph of g gis a horizontal compression by a factor of 1 5 1 5 of the graph of f. f.

59.

The graph of g gis a horizontal stretch by a factor of 3 of the graph of f. f.

61.

The graph of g gis a horizontal reflection across the y y-axis and a vertical stretch by a factor of 3 of the graph of f. f.

63.

g(x)=|4x| g(x)=|4x|

65.

g(x)= 1 3 (x+2) 2 3 g(x)= 1 3 (x+2) 2 3

67.

g(x)= 1 2 (x-5) 2 +1 g(x)= 1 2 (x-5) 2 +1

69.

The graph of the function f(x)= x 2 f(x)= x 2 is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

Graph of a parabola.
71.

The graph of f(x)=|x| f(x)=|x|is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

Graph of an absolute function.
73.

The graph of the function f(x)= x 3 f(x)= x 3 is compressed vertically by a factor of 1 2 . 1 2 .

Graph of a cubic function.
75.

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

Graph of a cubic function.
77.

The graph of f(x)= x f(x)= x is shifted right 4 units and then reflected across the vertical line x=4. x=4.

Graph of a square root function.
79.
Graph of a polynomial.
81.
Graph of a polynomial.

1.6 Section Exercises

1.

Isolate the absolute value term so that the equation is of the form |A|=B. |A|=B.Form one equation by setting the expression inside the absolute value symbol, A, A,equal to the expression on the other side of the equation, B. B.Form a second equation by setting A Aequal to the opposite of the expression on the other side of the equation, B. B.Solve each equation for the variable.

3.

The graph of the absolute value function does not cross the x x-axis, so the graph is either completely above or completely below the x x-axis.

5.

First determine the boundary points by finding the solution(s) of the equation. Use the boundary points to form possible solution intervals. Choose a test value in each interval to determine which values satisfy the inequality.

7.

| x+4 |= 1 2 | x+4 |= 1 2

9.

|f(x)8|<0.03 |f(x)8|<0.03

11.

{ 1,11 } { 1,11 }

13.

{ 9 4 , 13 4 } { 9 4 , 13 4 }

15.

{ 10 3 , 20 3 } { 10 3 , 20 3 }

17.

{ 11 5 , 29 5 } { 11 5 , 29 5 }

19.

{ 5 2 , 7 2 } { 5 2 , 7 2 }

21.

No solution

23.

{ 57,27 } { 57,27 }

25.

( 0,8 );( 6,0 ),( 4,0 ) ( 0,8 );( 6,0 ),( 4,0 )

27.

( 0,7 ); ( 0,7 );no x x-intercepts

29.

(,8)(12,) (,8)(12,)

31.

4 3 x4 4 3 x4

33.

( , 8 3 ][ 6, ) ( , 8 3 ][ 6, )

35.

( , 8 3 ][ 16, ) ( , 8 3 ][ 16, )

37.
Graph of an absolute function with points at (-1, 2), (0, 1), (1, 0), (2, 1), and (3, 2).
39.
Graph of an absolute function with points at (-2, 3), (-1, 2), (0, 1), (1, 2), and (2, 3).
41.
Graph of an absolute function.
43.
Graph of an absolute function.
45.
Graph of an absolute function.
47.
Graph of an absolute function.
49.
Graph of an absolute function.
51.
Graph of an absolute function.
53.

range: [ 0,20 ] [ 0,20 ]

Graph of an absolute function.
55.

x- x- intercepts:

Graph of an absolute function.
57.

(,) (,)

59.

There is no solution for a athat will keep the function from having a y y-intercept. The absolute value function always crosses the y y-intercept when x=0. x=0.

61.

| p0.08 |0.015 | p0.08 |0.015

63.

| x5.0 |0.01 | x5.0 |0.01

1.7 Section Exercises

1.

Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that y y-values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no y y-values repeat and the function is one-to-one.

3.

Yes. For example, f(x)= 1 x f(x)= 1 x is its own inverse.

5.

Given a function y=f(x), y=f(x),solve for x xin terms of y. y.Interchange the x xand y. y.Solve the new equation for y. y.The expression for y yis the inverse, y= f 1 (x). y= f 1 (x).

7.

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

9.

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

11.

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

13.

domain of f(x):[7,); f 1 (x)= x 7 f(x):[7,); f 1 (x)= x 7

15.

domain of f(x):[0,); f 1 (x)= x+5 f(x):[0,); f 1 (x)= x+5

16.

a.  f(g(x))=x  f(g(x))=xand g(f(x))=x. g(f(x))=x.b. This tells us that f fand g gare inverse functions

17.

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

19.

one-to-one

21.

one-to-one

23.

not one-to-one

25.

3 3

27.

2 2

29.
Graph of a square root function and its inverse.
31.

[ 2,10 ] [ 2,10 ]

33.

6 6

35.

4 4

37.

0 0

39.

1 1

41.
x x 1 4 7 12 16
f 1 (x) f 1 (x) 3 6 9 13 14
43.

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

Graph of a cubic function and its inverse.
45.

f 1 (x)= 5 9 ( x32 ). f 1 (x)= 5 9 ( x32 ).Given the Fahrenheit temperature, x, x,this formula allows you to calculate the Celsius temperature.

47.

t(d)= d 50 , t(d)= d 50 , t(180)= 180 50 . t(180)= 180 50 .The time for the car to travel 180 miles is 3.6 hours.

Review Exercises

1.

function

3.

not a function

5.

f(3)=27; f(3)=27; f(2)=2; f(2)=2; f(a)=2 a 2 3a; f(a)=2 a 2 3a;
f(a)=2 a 2 3a; f(a)=2 a 2 3a; f(a+h)=2 a 2 +3a4ah+3h2 h 2 f(a+h)=2 a 2 +3a4ah+3h2 h 2

7.

one-to-one

9.

function

11.

function

13.
Graph of f(x).
15.

2 2

17.

x=1.8  x=1.8 or  or x=1.8  or x=1.8

19.

64+80a16 a 2 1+a =16a+64 64+80a16 a 2 1+a =16a+64

21.

( ,2 )( 2,6 )( 6, ) ( ,2 )( 2,6 )( 6, )

23.
Graph of f(x).
25.

31 31

27.

increasing ( 2, ); ( 2, ); decreasing (,2) (,2)

29.

increasing ( 3,1 ); ( 3,1 ); constant (,3)( 1, ) (,3)( 1, )

31.

local minimum ( 2,3 ); ( 2,3 );local maximum ( 1,3 ) ( 1,3 )

33.

Absolute Maximum: 10

35.

( fg )(x)=1718x;( gf )(x)=718x ( fg )(x)=1718x;( gf )(x)=718x

37.

( fg )(x)= 1 x +2 ;( gf )(x)= 1 x+2 ( fg )(x)= 1 x +2 ;( gf )(x)= 1 x+2

39.

(fg)(x)= 1+x 1+4x , x0, x 1 4 (fg)(x)= 1+x 1+4x , x0, x 1 4

41.

( fg )(x)= 1 x ,x>0 ( fg )(x)= 1 x ,x>0

43.

sample: g(x)= 2x1 3x+4 ;f(x)= x g(x)= 2x1 3x+4 ;f(x)= x

45.
Graph of f(x)
47.
Graph of f(x)
49.
Graph of f(x)
51.
Graph of f(x)
53.
Graph of a half circle.
55.

f(x)=| x3 | f(x)=| x3 |

57.

even

59.

odd

61.

even

63.

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

65.

f(x)=3| x3 |+3 f(x)=3| x3 |+3

67.
Graph of f(x).
69.

x=22, x=14 x=22, x=14

71.

( 5 3 ,3 ) ( 5 3 ,3 )

73.

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

77.

The function is one-to-one.

78.

The function is not one-to-one.

79.

5 5

Practice Test

1.

The relation is a function.

3.

−16

5.

The graph is a parabola and the graph fails the horizontal line test.

7.

2 a 2 a 2 a 2 a

9.

2(a+b)+1 2(a+b)+1

11.

2 2

13.
Graph of f(x).
15.

even even

17.

odd odd

19.

x=7 x=7and x=10 x=10

21.

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

23.

(,1.1) and (1.1,) (,1.1) and (1.1,)

25.

( 1.1,0.9 ) ( 1.1,0.9 )

27.

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

29.

f(x)={ | x |ifx2 3ifx>2 f(x)={ | x |ifx2 3ifx>2

31.

x=2 x=2

33.

yes

35.

f 1 (x)= x11 2 f 1 (x)= x11 2

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