SECTION 1.1 An Introduction to Functions

Chapter 1 A Review of Functions

Section 1.1:

An Introduction to Functions

 Definition of a Function and Evaluating a Function  Domain and Range of a Function

Definition of a Function and Evaluating a Function Definition:

MATH 1330 Precalculus

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CHAPTER 1 A Review of Functions

Defining a Function by an Equation in the Variables x and y:

Example:

Solution:

The Function Notation:

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University of Houston Department of Mathematics

SECTION 1.1 An Introduction to Functions

Evaluating a Function:

Example:

Solution:

MATH 1330 Precalculus

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CHAPTER 1 A Review of Functions

Example:

Solution:

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University of Houston Department of Mathematics

SECTION 1.1 An Introduction to Functions

Difference Quotients:

Example:

Solution:

Additional Example 1:

Solution:

MATH 1330 Precalculus

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CHAPTER 1 A Review of Functions

Additional Example 2:

Solution:

Additional Example 3:

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University of Houston Department of Mathematics

SECTION 1.1 An Introduction to Functions Solution:

Additional Example 4:

Solution:

MATH 1330 Precalculus

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CHAPTER 1 A Review of Functions Additional Example 5:

Solution:

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University of Houston Department of Mathematics

SECTION 1.1 An Introduction to Functions Additional Example 6:

Solution:

MATH 1330 Precalculus

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CHAPTER 1 A Review of Functions Additional Example 7:

Solution:

Domain and Range of a Function Review of Interval Notation:

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University of Houston Department of Mathematics

SECTION 1.1 An Introduction to Functions

Finding the Domain of a Function:

Example:

Solution:

MATH 1330 Precalculus

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CHAPTER 1 A Review of Functions

Example:

Solution:

Finding the Range of a Function:

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University of Houston Department of Mathematics

SECTION 1.1 An Introduction to Functions Example:

Solution:

Additional Example 1:

Solution:

MATH 1330 Precalculus

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CHAPTER 1 A Review of Functions Additional Example 2:

Solution:

Additional Example 3:

Solution:

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University of Houston Department of Mathematics

SECTION 1.1 An Introduction to Functions

Additional Example 4:

Solution:

Additional Example 5:

Solution:

MATH 1330 Precalculus

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CHAPTER 1 A Review of Functions

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University of Houston Department of Mathematics

SECTION 1.1 An Introduction to Functions

MATH 1330 Precalculus

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CHAPTER 1 A Review of Functions

Additional Example 6:

Solution:

Additional Example 7:

Solution:

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University of Houston Department of Mathematics

SECTION 1.1 An Introduction to Functions

MATH 1330 Precalculus

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Exercise Set 1.1: An Introduction to Functions For each of the examples below, determine whether the mapping makes sense within the context of the given situation, and then state whether or not the mapping represents a function. 1.

Erik conducts a science experiment and maps the temperature outside his kitchen window at various times during the morning. 57 9

62

10

7.

Divide by 7, then add 4

8.

Multiply by 2, then square

9.

Take the square root, then subtract 6

10. Add 4, square, then subtract 2

65 o

Time

2.

Express each of the following rules in function notation. (For example, “Subtract 3, then square” would be written as f ( x )  ( x  3)2 .)

Temp. ( F)

Dr. Kim counts the number of people in attendance at various times during his lecture this afternoon. 1

85

2

Find the domain of each of the following functions. Then express your answer in interval notation. 11. f ( x) 

5 x3

12. f ( x) 

x6 x 1

87

3

13. g ( x) 

Time

# of People

State whether or not each of the following mappings represents a function. If a mapping is a function, then identify its domain and range.

14. f ( x) 

15. f ( x) 

3. 7 9 -3 A

0 5 4

16. g ( x) 

x4 x2  9 3x  1 x2  4 x2  6 x  5 x 2  11x  28

3x  15 x  8 x  20 2

B

17. f (t )  t 4. -6

8 4 -7

A

B

9

18. h( x)  3 x 19. g ( x)  5 x 20. h(t )  4 t

5.

9 -2

-6

21. f ( x)  x  5

1 A

6.

B

0 8

20

23. F ( x) 

3  2x x4

24. G ( x) 

x3 x7

2

4 A

22. g ( x)  x  7

B

University of Houston Department of Mathematics

Exercise Set 1.1: An Introduction to Functions 25. f ( x)  3 x  5

35. (a)

f (t )  t

(b) g (t )  9  t 26. g ( x)  3 x2  x  6 27. h(t )  3

t 2  7t  8 t 5

28. f ( x)  5

2x  9 4x  7

29. f (t )  t 2  10t  24

(c) h(t )  9  t 36. (a)

f ( x)  x

(b) g ( x)  x  1 (c) h( x)  x  1 37. (a) f ( x)  x 2  3 (b) g ( x)  x 2  3  4

30. g (t )  t 2  5t  14

Find the domain and range of each of the following functions. Express answers in interval notation. 31. (a)

f ( x)  x

(b) g ( x)  x  6

(c) h( x)  2 x 2  3  5

38. (a)

f (t )  t 2  6

(b) g (t )  t 2  6  7 (c) h(t )   13 t 2  6  8

(c) h( x)  x  6 (d) p( x)  x  6  3 32. (a) f (t )  3  t

39. g (t )  2 x  7  5 40. h(t )  6  t  1

(b) g (t )  3  t (c) h(t )  3  t (d) p(t )  3  t  7 33. (a)

42. g ( x)  4 8  3x  2

f ( x)  x 2  4

(b) g ( x)  4  x 2 (c) h( x)  x 2  4 (d) p ( x)  x 2  4 (e) q ( x)  4  x 2 (f) r ( x)  x 2  4 34. (a)

41. f ( x)  3 5x  6  4

Find the domain and range of each of the following functions. Express answers in interval notation. (Hint: When finding the range, first solve for x.) 3 x2 x5 (b) g ( x)  x2

43. (a)

f ( x) 

f (t )  25  t 2

(b) g (t )  t 2  25 (c) h(t )  25  t 2 (d) p(t )  25  t 2

4 x3 5x  2 (b) g ( x)  x 3

44. (a) f ( x) 

(e) q (t )  t 2  25 (f) r (t )  25  t 2

MATH 1330 Precalculus

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Exercise Set 1.1: An Introduction to Functions 3x 2 , if x  0   53. If f ( x)  4, if 0  x  2 , find:  x  5, if x  2  

Evaluate the following. 45. If f ( x)  5x  4 , find:

 

f (3), f  12 , f (a), f (a  3), f (a)  3, f (a)  f (3)

f (0), f (6), f 2, f (1), f (4), f

32 

46. If f ( x)  3x  1 , find:

 

f (5), f (8), f  74 , f (t )  2, f (t  2), f (t )  f (2)

47. If g ( x)  x 2  3x  4 , find:

4 x  7, if x  1  54. If f ( x)   x 2  6, if - 1  x  3 , find:  7, if x  3 

 

f (0), f (4), f  1, f (3), f (6), f  53

 , g ( x  5), g  , g (3a), 3g (a)

g (0), g 

1 4

1 a

48. If h(t )  t 2  2t  5 , find: h(1), h

32 , h(c  6),  h( x), h(2x), 2h( x)

Determine whether each of the following equations defines y as a function of x. (Do not graph.) 55. 3x  5 y  8

49. If f ( x) 

2 x , find: x3

f (7), f (0), f

53 ,

56. x  3  y 2

f (t ), f (t 2  3)

57. x 2  y  3 58. 3x 4  2 xy  5 x

x2  x , find: 50. If f ( x)  x4

59. 7 x  y 4  5

 

f (2), f (5), f  74 , f (3 p), f ( p3 )  2

61. x3 y  2 y  6

2 x  5, if x  4 , find: 3  x 2 , if x  4

51. If f ( x)  

f (6), f (2), f  3, f (0), f (4), f

 9 2

 x 2  4 x, if x  2 52. If f ( x)   , find: 7  2 x, if x  2

 

f (5), f (3), f 0, f (1), f (2), f  10 3

22

60. 3x 2  y 2  16

62.

x  3y  5

63. x3  5 y 64. y 3  7 x 65.

y 2  x

66.

x  3y  4

University of Houston Department of Mathematics

Exercise Set 1.1: An Introduction to Functions For each of the following problems: (a) Find f ( x  h) . (b) Find the difference quotient (Assume that h  0 .)

f ( x  h)  f ( x ) . h

67. f ( x)  7 x  4 68. f ( x)  5  3x 69. f ( x)  x 2  5 x  2 70. f ( x)  x 2  3x  8 71. f ( x)  8 72. f ( x)  6 73. f  x  

1 x

74. f  x  

1 x3

MATH 1330 Precalculus

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