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:
2
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:
4
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:
10
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:
12
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:
14
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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