Introduction to Functions Section 2.6

Introduction to Functions Section 2.6
miL96478_ch02_191-202.qxd
9/19/09
5:10 AM
Page 191
INTERM—AIE
Section 2.6
Introduction to Functions
Introduction to Functions
Section 2.6
1. Definition of a Function
Concepts
In this section, we introduce a special type of relation called a function.
1.
2.
3.
4.
DEFINITION Function
Given a relation in x and y, we say “y is a function of x” if, for each element
x in the domain, there is exactly one value of y in the range.
Definition of a Function
Vertical Line Test
Function Notation
Finding Function Values
from a Graph
5. Domain of a Function
Note: This means that no two ordered pairs may have the same first coordinate and different second coordinates.
To understand the difference between a relation that is a function and a relation
that is not a function, consider Example 1.
Example 1
Determining Whether a Relation Is a Function
Classroom Examples: p. 198,
Exercises 6 and 10
Determine which of the relations define y as a function of x.
a.
b.
x
y
2
3
4
⫺1
3
⫺2
1
c.
x
y
x
1
2
1
2
⫺1
2
3
4
3
Solution:
a. This relation is defined by the set of ordered pairs
same x
511, 32, 11, 42, 12, ⫺12, 13, ⫺226
different y-values
When x ⫽ 1, there are two possible range elements: y ⫽ 3 and y ⫽ 4.
Therefore, this relation is not a function.
b. This relation is defined by the set of ordered pairs 511, 42, 12, ⫺12, 13, 226.
Notice that no two ordered pairs have the same value of x but different
values of y. Therefore, this relation is a function.
c. This relation is defined by the set of ordered pairs 511, 42, 12, 42, 13, 426.
Notice that no two ordered pairs have the same value of x but different
values of y. Therefore, this relation is a function.
y
4
191
miL96478_ch02_191-202.qxd
9/19/09
5:11 AM
Page 192
INTERM—AIE
192
Chapter 2 Linear Equations in Two Variables and Functions
Instructor Note: Explain that a
function can have the same
y-coordinate with different
x-coordinates, but not have the
same x-coordinate with different
y-coordinates.
Skill Practice Determine if the relations define y as a function of x.
1.
x
y
2
12
6
13
7
10
2. {(4, 2), (⫺5, 4), (0, 0), (8, 4)}
3. {(⫺1, 6), (8, 9), (⫺1, 4), (⫺3, 10)}
2. Vertical Line Test
y
5
4
3
2
1
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
⫺1
⫺2
Points
align
vertically
1 2
(4, 2)
3 4 5
x
(4, ⫺2)
⫺3
⫺4
⫺5
Figure 2-32
A relation that is not a function has at least one domain element x paired with more
than one range value y. For example, the set {(4, 2), (4, ⫺2)} does not define a function because two different y-values correspond to the same x. These two points are
aligned vertically in the xy-plane, and a vertical line drawn through one point also intersects the other point (see Figure 2-32). If a vertical line drawn through a graph of
a relation intersects the graph in more than one point, the relation cannot be a function.
This idea is stated formally as the vertical line test.
PROCEDURE The Vertical Line Test
Consider a relation defined by a set of points (x, y) in a rectangular coordinate system. The graph defines y as a function of x if no vertical line intersects
the graph in more than one point.
The vertical line test can be demonstrated by graphing the ordered pairs from
the relations in Example 1.
a. 511, 32, 11, 42, 12, ⫺12, 13, ⫺226
b. 511, 42, 12, ⫺12, 13, 226
y
y
5
5
Intersects
more than
once
4
3
2
4
3
2
1
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
⫺1
⫺2
1
1
3 4 5
⫺3
⫺4
⫺5
Example 2
1. Yes
2. Yes
3. No
1 2
3 4 5
x
⫺5
Function
No vertical line
intersects more than once.
Using the Vertical Line Test
Use the vertical line test to determine whether the relations define y as a function of x.
a.
b.
y
y
x
Answers
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
⫺1
⫺2
⫺3
⫺4
Not a Function
A vertical line intersects
in more than one point.
Classroom Examples: p. 199,
Exercises 12 and 16
x
x
miL96478_ch02_191-202.qxd
7/21/09
12:11 PM
Page 193
INTERM—AIE
Section 2.6
Introduction to Functions
193
Solution:
a.
b.
y
y
x
x
Not a Function
A vertical line intersects
in more than one point.
Function
No vertical line intersects
in more than one point.
Skill Practice Use the vertical line test to determine whether the relations
define y as a function of x.
4.
5.
y
y
x
x
3. Function Notation
A function is defined as a relation with the added restriction that each value in the
domain must have only one corresponding y value in the range. In mathematics,
functions are often given by rules or equations to define the relationship between
two or more variables. For example, the equation y 2x defines the set of ordered
pairs such that the y value is twice the x value.
When a function is defined by an equation, we often use function notation. For
example, the equation y 2x may be written in function notation as
f 1x2 2x
•
•
•
f is the name of the function.
x is an input value from the domain of the function.
f (x) is the function value (y value) corresponding to x.
The notation f (x) is read as “f of x” or “the value of the function f at x.”
A function may be evaluated at different values of x by substituting x-values
from the domain into the function. For example, to evaluate the function defined
by f1x2 2x at x 5, substitute x 5 into the function.
Avoiding Mistakes
Be sure to note that f (x ) is not f ⴢ x.
f 1x2 2x
f 152 2152
f 152 10
TIP: f 152 10 can be
written as the ordered pair
(5, 10).
Answers
4. Yes
5. No
miL96478_ch02_191-202.qxd
7/21/09
12:11 PM
Page 194
INTERM—AIE
194
Chapter 2 Linear Equations in Two Variables and Functions
Thus, when x 5, the corresponding function value is 10. We say:
f of 5 is 10.
f at 5 is 10.
f evaluated at 5 is 10.
•
•
•
The names of functions are often given by either lowercase or uppercase letters,
such as f, g, h, p, K, and M. The input variable may also be a letter other than x. For
example, the function y P(t) might represent population as a function of time.
Evaluating a Function
Example 3
Classroom Examples: p. 199,
Exercises 18 and 22
Given the function defined by g1x2 12x 1, find the function values.
a. g102
b. g122
c. g142
d. g122
Solution:
1
a. g1x2 x 1
2
g102 1
b. g1x2 x 1
2
1
102 1
2
g122 01
11
1
0
We say that “g of 0 is 1.”
This is equivalent to the ordered
pair 10, 12.
We say that “g of 2 is 0.”
This is equivalent to the ordered
pair 12, 02.
1
c. g1x2 x 1
2
g142 d.
1
142 1
2
21
1 1
1
2
We say that “g of 2 is 2.”
This is equivalent to the ordered
pair 12, 22.
Skill Practice Given the function defined by f (x) 2x 3, find the
function values.
y
5
4
g(x) 12 x 1
3
2
1
6. f (1)
7. f (0)
8. f (3)
1
9. f a b
2
(4, 1)
5 4 3 2 1
1 (2, 0)
1
(0, 1)
2
5
(2, 2)
4
5
Figure 2-33
Answers
7. 3
1
g1x2 x 1
2
1
g122 122 1
2
We say that “g of 4 is 1.”
This is equivalent to the ordered
pair 14, 12.
6. 5
1
122 1
2
8. 3
9. 4
x
Notice that g102, g122, g142, and g122 correspond to the ordered pairs 10, 12,
12, 02, 14, 12, and 12, 22. In the graph, these points “line up.” The graph of all
ordered pairs defined by this function is a line with a slope of 12 and y-intercept of
10, 12 (Figure 2-33). This should not be surprising because the function defined by
g1x2 12x 1 is equivalent to y 12x 1.
miL96478_ch02_191-202.qxd
9/19/09
5:12 AM
Page 195
INTERM—AIE
Section 2.6
Introduction to Functions
Calculator Connections
The values of g1x2 in Example 3 can be found
using a Table feature.
Y1 ⫽ 12x ⫺ 1
Function values can also be evaluated by
using a Value (or Eval) feature. The value of
g142 is shown here.
A function may be evaluated at numerical values or at algebraic expressions,
as shown in Example 4.
Evaluating Functions
Example 4
Given the functions defined by f1x2 ⫽ x ⫺ 2x and g1x2 ⫽ 3x ⫹ 5, find the
function values.
2
b. g1w ⫹ 42
a. f1t2
Classroom Examples: p. 199,
Exercises 24 and 38
c. f 1⫺t2
Solution:
a. f1x2 ⫽ x2 ⫺ 2x
f1t2 ⫽ 1t2 2 ⫺ 21t2
⫽ t2 ⫺ 2t
Substitute x ⫽ t for all values of x in the
function.
Simplify.
g1x2 ⫽ 3x ⫹ 5
b.
g1w ⫹ 42 ⫽ 31w ⫹ 42 ⫹ 5
Substitute x ⫽ w ⫹ 4 for all values of x in
the function.
⫽ 3w ⫹ 12 ⫹ 5
⫽ 3w ⫹ 17
c.
f 1x2 ⫽ x2 ⫺ 2x
f 1⫺t2 ⫽ 1⫺t2 ⫺ 21⫺t2
Simplify.
Substitute ⫺t for x.
2
⫽ t 2 ⫹ 2t
Simplify.
Skill Practice Given the function defined by g(x ) ⫽ 4x ⫺ 3, find the
function values.
10. g(a)
11. g(x ⫹ h)
12. g (⫺x )
Answers
10. 4a ⫺ 3
12. ⫺4x ⫺ 3
11. 4x ⫹ 4h ⫺ 3
195
miL96478_ch02_191-202.qxd
7/21/09
12:12 PM
Page 196
INTERM—AIE
196
Chapter 2 Linear Equations in Two Variables and Functions
4. Finding Function Values from a Graph
We can find function values by looking at a graph of the function. The value of f(a)
refers to the y-coordinate of a point with x-coordinate a.
Classroom Example: p. 200,
Exercise 58
Example 5
Finding Function Values from a Graph
Consider the function pictured in Figure 2-34.
h(x)
5
a. Find h112 .
y h(x) 4
3
2
1
b. Find h122 .
c. Find h152 .
5 4 3 2 1
1
2
d. For what value of x is h1x2 3?
1
2
3 4
5
x
3
e. For what values of x is h1x2 0?
4
5
Solution:
Figure 2-34
a. h112 2
This corresponds to the ordered pair (1, 2).
b. h122 1
This corresponds to the ordered pair (2, 1).
c. h152 is not defined.
The value 5 is not in the domain.
d. h1x2 3
for x 4
This corresponds to the ordered
pair (4, 3).
e. h1x2 0
for x 3 and x 4
These are the ordered pairs (3, 0)
and (4, 0).
Skill Practice Refer to the function graphed here.
13.
14.
15.
16.
17.
Find f (0).
Find f (2).
Find f (5).
For what value(s) of x is f (x ) 0?
For what value(s) of x is f (x ) 4?
y
5
4
3
2
1
5 4 3 2 1
1
2
y f(x)
1
2
3 4
5
x
3
4
5
5. Domain of a Function
A function is a relation, and it is often necessary to determine its domain and range.
Consider a function defined by the equation y f 1x2 . The domain of f is the set of all
x-values that when substituted into the function produce a real number. The range of
f is the set of all y-values corresponding to the values of x in the domain.
To find the domain of a function defined by y f 1x2, keep these guidelines in
mind.
•
•
Answers
13.
15.
16.
17.
3
14. 1
not defined
x 4 and x 4
x 5
Exclude values of x that make the denominator of a fraction zero.
Exclude values of x that make the expression within a square root negative.
miL96478_ch02_191-202.qxd
9/19/09
5:21 AM
Page 197
INTERM—AIE
Section 2.6
Finding the Domain of a Function
Example 6
Classroom Examples: p. 201,
Exercises 74, 80, and 88
Write the domain in interval notation.
a. f1x2 ⫽
x⫹7
2x ⫺ 1
c. k1t2 ⫽ 1t ⫹ 4
b. h1x2 ⫽
197
Introduction to Functions
x⫺4
x2 ⫹ 9
Instructor Note: Example 6 shows
students that they can find the
domain of a function without needing
its graph. If you want to emphasize
the graphical relationship, show the
students the graphs of these
functions in class. Or have the
students graph the functions on a
graphing calculator.
d. g1t2 ⫽ t2 ⫺ 3t
Solution:
a. f 1x2 ⫽ 2xx ⫹⫺ 71 will not be a real number when the denominator is zero, that
is, when
2x ⫺ 1 ⫽ 0
2x ⫽ 1
x⫽
1
2
The value x ⫽ 12 must be excluded from the domain.
1
2
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
(((
0
1
2
3
4
5
6
1
1
Interval notation: a⫺⬁, b ´ a , ⬁b
2
2
b. For h1x2 ⫽ xx2 ⫺⫹ 49 the quantity x2 is greater than or equal to 0 for all real
numbers x, and the number 9 is positive. The sum x2 ⫹ 9 must be positive
for all real numbers x. The denominator will never be zero; therefore, the
domain is the set of all real numbers.
Interval notation: 1⫺⬁, ⬁2
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
4
5
6
c. The function defined by k1t2 ⫽ 1t ⫹ 4 will not be a real number when the
radicand is negative. The domain is the set of all t values that make the
radicand greater than or equal to zero:
t⫹4ⱖ0
t ⱖ ⫺4
Interval notation: 3⫺4, ⬁ 2
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
4
5
6
d. The function defined by g1t2 ⫽ t2 ⫺ 3t has no restrictions on its domain
because any real number substituted for t will produce a real number. The
domain is the set of all real numbers.
Interval notation: 1⫺⬁, ⬁2
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
4
5
6
Skill Practice Write the domain in interval notation.
18. f 1x2 ⫽
2x ⫹ 1
x⫺9
20. g 1x2 ⫽ 1x ⫺ 2
19. k 1x 2 ⫽
⫺5
4x 2 ⫹ 1
21. h 1x2 ⫽ x ⫹ 6
Answers
18. 1⫺⬁, 92 ´ 19, ⬁2
20. 32, ⬁2
19. 1⫺⬁, ⬁ 2
21. 1⫺⬁, ⬁2
miL96478_ch02_191-202.qxd
8/10/09
8:18 PM
Page 198
INTERM—AIE
198
Chapter 2 Linear Equations in Two Variables and Functions
Section 2.6 Practice Exercises
Boost your GRADE at
ALEKS.com!
• Practice Problems
• Self-Tests
• NetTutor
For additional exercises, see Classroom Activities
2.6A–2.6D in the Instructor’s Resource Manual
at www.mhhe.com/moh.
• e-Professors
• Videos
Study Skills Exercises
1. Look back over your notes for this chapter. Have you highlighted the important topics? Have you underlined
the key terms? Have you indicated the places where you are having trouble? If you find that you have
problems with a particular topic, write a question that you can ask your instructor either in class or in the
instructor’s office.
2. Define the key terms.
a. Function
b. Function notation
c. Domain
d. Range
e. Vertical line test
Review Exercises
For Exercises 3–4, a. write the relation as a set of ordered pairs, b. identify the domain, and c. identify the range.
3.
Parent, x
Child, y
Kevin
Kayla
Kevin
Kira
Kathleen
Katie
Kathleen
y
4.
a. {(⫺2,⫺4), (⫺1, ⫺1), (0, 0), (1, ⫺1), (2, ⫺4)}
b. {⫺2, ⫺1, 0, 1, 2}
c. {⫺4, ⫺1, 0}
5
4
3
2
1
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
⫺1
⫺2
Kira
a. {(Kevin, Kayla), (Kevin, Kira),
(Kathleen, Katie), (Kathleen, Kira)}
b. {Kevin, Kathleen}
c. {Kayla, Katie, Kira}
1
2
3 4 5
x
⫺3
⫺4
⫺5
Concept 1: Definition of a Function
For Exercises 5–10, determine if the relation defines y as a function of x. (See Example 1.)
5.
x
6.
y
x
y
⫺2
1
⫺2
1
2
2
2
2
0
8
0
8
7.
u
21
w
23
x
24
y
25
z
26
Not a function
Function
Not a function
9. 511, 22, 13, 42, 15, 42, 1⫺9, 326
8.
3
10
4
Function
1
1
10. e10, ⫺1.12, a , 8b, 11.1, 82, a4, bf
2
2
Function
12
5
6
Function
Writing
Translating Expression
Geometry
Scientific Calculator
Video
miL96478_ch02_191-202.qxd
7/21/09
12:12 PM
Page 199
INTERM—AIE
Section 2.6
199
Introduction to Functions
Concept 2: Vertical Line Test
For Exercises 11–16, use the vertical line test to determine whether the relation defines y as a function of x.
(See Example 2.)
11.
12.
y
13.
y
x
x
x
Function
Not a function
14.
y
15.
y
Function
16.
y
x
y
x
Not a function
x
Not a function
Not a function
Concept 3: Function Notation
Consider the functions defined by f1x2 6x 2, g1x2 x2 4x 1,
Exercises 17–48, find the following. (See Examples 3–4.)
h1x2 7, and k1x2 0 x 2 0 . For
17. g122
11
18. k122
0
19. g102
1
20. h102
7
21. k102
2
22. f 102
2
23. f 1t2
6t 2
24. g1a2
a2 4a 1
25. h1u2
7
26. k1v2
冟v 2冟
27. g132
29. k122
33. g12x2
30. f 162
4
4x2 8x 1
37. h1a b2
7
38
28. h152
4
31. f 1x 12
32. h1x 12
6x 4
36. g1a2 2
34. k1x 32
冟 x 5冟
35. g1p2
38. f 1x h2
6x 6h 2
39. f 1a2
6a 2
40. g1b2
1
1
44. g a b
4
p2 4p 1
41. k1c2
冟c 2冟
42. h1x2
7
1
43. f a b
2
1
45. h a b
7
7
3
46. k a b
2
1
2
47. f 12.82
7
18.8
7
a4 4a2 1
b2 4b 1
48. k15.42
1
16
7.4
Consider the functions p 51 12, 62, 12, 72, 11, 02, 13, 2p26 and q 516, 42, 12, 52, 1 34, 15 2, 10, 926. For Exercises
49–56, find the function values.
49. p122
7
50. p112
53. q122
5
3
54. qa b
4
Writing
0
1
5
Translating Expression
51. p132
2p
1
52. pa b
2
55. q162
4
56. q102
Geometry
Scientific Calculator
Video
6
9
miL96478_ch02_191-202.qxd
7/21/09
12:12 PM
Page 200
INTERM—AIE
200
Chapter 2 Linear Equations in Two Variables and Functions
Concept 4: Finding Function Values from a Graph
57. The graph of y f 1x2 is given. (See Example 5.)
a. Find f102 .
3
b. Find f132.
1
c. Find f122.
y
5
1
d. For what value(s) of x is f1x2 3?
e. For what value(s) of x is f1x2 3?
f. Write the domain of f.
g. Write the range of f.
5 4 3 2 1
1
2
x 3
1
c. Find g142.
1
1, 34
x 3, x 5
f. Write the domain of g.
g. Write the range of g.
H142 is not defined because 4 is not in the domain of H.
4
5
4
3
2
1
5 4 3 2 1
1
2
3
d. For what value(s) of x is H1x2 3?
x 3 and x 2
e. For what value(s) of x is H1x2 2?
3 4
5
x
4
5
3 4, 42
3 2, 52
60. The graph of y K1x2 is given.
y
5
4
3 y K(x)
2
1
1
b. Find K152.
K152 is not defined because 5 is not in the domain of K.
0
c. Find K112.
d. For what value(s) of x is K1x2 0?
x 1, x 1, and x 3
e. For what value(s) of x is K1x2 3?
f. Write the domain of K.
2
x
y
y H(x)
c. Find H132.
g. Write the range of H.
1
4 5
1, 3 4
b. Find H142.
f. Write the domain of H.
2 3
4
5
33, 2
3
All x on the interval 冤2, 1冥
1
2
3
59. The graph of y H1x2 is given.
Writing
x
y g(x)
5 4 3 2 1
1
e. For what value(s) of x is g1x2 0?
g. Write the range of K.
5
y
5
4
3
2
1
x2
a. Find K102.
3 4
1, 5 4
d. For what value(s) of x is g1x2 3?
a. Find H132.
2
4
5
2
b. Find g112.
1
3
x 0, x 2
58. The graph of y g1x2 is given.
a. Find g112.
y f(x)
4
3
2
1
x 4
5 4 3 2 1
1
2
3
1
2
3 4
5
x
4
5
15, 5 4
冤1, 42
Translating Expression
Geometry
Scientific Calculator
Video
miL96478_ch02_191-202.qxd
9/19/09
5:36 AM
Page 201
INTERM—AIE
Section 2.6
201
Introduction to Functions
For Exercises 61–70, refer to the functions y ⫽ f 1x2 and y ⫽ g1x2 , defined as follows:
f ⫽ 51⫺3, 52, 1⫺7, ⫺32, 1⫺32, 42, 11.2, 526
g ⫽ 510, 62, 12, 62, 16, 02, 11, 026
5⫺3, ⫺7, ⫺32 , 1.26
61. Identify the domain of f.
56, 06
63. Identify the range of g.
64. Identify the domain of g.
65. For what value(s) of x is f(x) ⫽ 5?
50, 2, 6, 16
66. For what value(s) of x is f(x) ⫽ ⫺3?
⫺3 and 1.2
67. For what value(s) of x is g(x) ⫽ 0?
69. Find f (⫺7).
55, ⫺3, 46
62. Identify the range of f.
68. For what value(s) of x is g(x) ⫽ 6?
6 and 1
⫺3
70. Find g(0).
Concept 5: Domain of a Function
71. Explain how to determine the domain of the function defined by f 1x2 ⫽
⫺7
0 and 2
6
x⫹6
.
x ⫺ 2 The domain is the set of all real numbers
for which the denominator is not zero. Set the denominator equal to zero, and solve the resulting equation. The solution(s) to the equation must be
excluded from the domain. In this case, setting x ⫺ 2 ⫽ 0 indicates that x ⫽ 2 must be excluded from the domain. The domain is 1⫺⬁, 22 ´ 12, ⬁2 .
The domain is the set of all
72. Explain how to determine the domain of the function defined by g1x2 ⫽ 1x ⫺ 3.
real numbers for which x ⫺ 3 is nonnegative. Set the quantity x ⫺ 3 ⱖ 0 and solve the inequality. The solution set for the inequality is the
domain of the function. Thus, the domain is 3 3, ⬁ 2 .
For Exercises 73–88, find the domain. Write the answers in interval notation. (See Example 6.)
73. k1x2 ⫽
x⫺3
x⫹6
1⫺⬁, ⫺62 ´ 1⫺6, ⬁2
77. h1p2 ⫽
1⫺⬁, ⬁ 2
p⫺4
p2 ⫹ 1
74. m1x2 ⫽
x⫺1
x⫺4
75. f1t2 ⫽
1⫺⬁, 42 ´ 14, ⬁ 2
78. n1p2 ⫽
1⫺⬁, ⬁ 2
5
t
76. g1t2 ⫽
1⫺⬁, 02 ´ 10, ⬁ 2
p⫹8
1⫺⬁, 02 ´ 10, ⬁2
79. h1t2 ⫽ 1t ⫹ 7
p2 ⫹ 2
t⫺7
t
80. k1t2 ⫽ 1t ⫺ 5
3 ⫺7, ⬁ 2
35, ⬁2
81. f1a2 ⫽ 1a ⫺ 3
82. g1a2 ⫽ 1a ⫹ 2
83. m1x2 ⫽ 11 ⫺ 2x
84. n1x2 ⫽ 112 ⫺ 6x
85. p1t2 ⫽ 2t ⫹ t ⫺ 1
86. q1t2 ⫽ t ⫹ t ⫺ 1
87. f1x2 ⫽ x ⫹ 6
88. g1x2 ⫽ 8x ⫺ p
3 3, ⬁2
1⫺⬁, ⬁ 2
2
3 ⫺2, ⬁ 2
1⫺⬁, ⬁ 2
1⫺⬁, 12 4
3
1⫺⬁, 2 4
1⫺⬁, ⬁ 2
1⫺⬁, ⬁ 2
Mixed Exercises
89. The height (in feet) of a ball that is dropped from an 80-ft building is given by h1t2 ⫽ ⫺16t2 ⫹ 80, where t is
the time in seconds after the ball is dropped.
a. Find h(1) and h(1.5)
h(1) ⫽ 64 and h(1.5) ⫽ 44
h(1) ⫽ 64 means that after
1 sec, the height of the ball is 64 ft. h(1.5) ⫽ 44 means that after 1.5 sec, the height of the ball is 44 ft.
b. Interpret the meaning of the function values found in part (a).
90. A ball is dropped from a 50-m building. The height (in meters) after t sec is given by h1t2 ⫽ ⫺4.9t2 ⫹ 50.
a. Find h(1) and h(1.5).
h(1) ⫽ 45.1 and h(1.5) ⫽ 38.975
h(1) ⫽ 45.1 means that after 1 sec, the height of the
ball is 45.1 m. h(1.5) ⫽ 38.975 means that after 1.5 sec, the height of the ball is 38.975 m.
b. Interpret the meaning of the function values found in part (a).
91. If Alicia rides a bike at an average speed of 11.5 mph, the distance that she rides can be represented by
d1t2 ⫽ 11.5t, where t is the time in hours.
a. Find d(1) and d(1.5).
d(1) ⫽ 11.5 and d(1.5) ⫽ 17.25
d(1) ⫽ 11.5 means that after 1 hr, the distance
d(1.5) ⫽ 17.25 means that after 1.5 hr, the distance traveled is 17.25 mi.
b. Interpret the meaning of the function values found in part (a).
traveled is 11.5 mi.
Writing
Translating Expression
Geometry
Scientific Calculator
Video
miL96478_ch02_191-202.qxd
9/19/09
5:38 AM
Page 202
INTERM—AIE
202
Chapter 2 Linear Equations in Two Variables and Functions
92. Brian’s score on an exam is a function of the number of hours he spends studying. The function defined by
P1x2 100x2
(x 0) indicates that he will achieve a score of P% if he studies for x hours.
50 x2
Evaluate P(0), P(5), P(10), P(15), P(20), and P(25) and confirm
the values on the graph. (Round to one decimal place.) Interpret P(25) in the context of this problem.
Student Score (Percent) as a
Function of Study Time
D
Percent
P102 0%
P152 ⬇ 33.3%
P1102 ⬇ 66.7%
P1152 ⬇ 81.8%
P1202 ⬇ 88.9%
P1252 ⬇ 92.6%. If Brian studies 25 hr he will get a score of 92.6%.
P(x)
100
90
80
70
60
50
40
30
20
10
0 A
0
F
E
C
B
5
10
15
20
Study Time (hr)
25
30
x
For Exercises 93–96, write a function defined by y f(x) subject to the conditions given.
93. The value of f(x) is three more than two times x.
94. The value of f(x) is four less than the square of x.
f (x) 2x 3
f(x) x2 4
95. The value of f(x) is ten less than the absolute
value of x. f(x) |x| 10
96. The value of f(x) is sixteen times the square root
of x. f(x) 16 1x
Expanding Your Skills
For Exercises 97–98, write the domain in interval notation.
97. q1x2 12, 2
2
98. p1x2 2x 2
14, 2
8
2x 4
Graphing Calculator Exercises
99. Graph k1t2 1t 5. Use the graph to support
your answer to Exercise 80.
Section 2.7
100. Graph h1t2 1t 7. Use the graph to support
your answer to Exercise 79.
Graphs of Functions
Concepts
1. Linear and Constant Functions
1. Linear and Constant
Functions
2. Graphs of Basic Functions
3. Definition of a Quadratic
Function
4. Finding the x- and
y-Intercepts of a Graph
Defined by y ⫽ f (x)
A function may be expressed as a mathematical equation that relates two or more variables. In this section,
we will look at several elementary functions.
We know from Section 2.1 that an equation in the
form y k, where k is a constant, is a horizontal line.
In function notation, this can be written as f1x2 k.
For example, the graph of the function defined by
f1x2 3 is a horizontal line, as shown in Figure 2-35.
We say that a function defined by f1x2 k is a
constant function because for any value of x, the function value is constant.
Writing
Translating Expression
Geometry
Scientific Calculator
Video
f(x)
5
f(x) 3
4
3
2
1
5 4 3 2 1
1
2
1
2
3
4
5
Figure 2-35
3 4
5
x
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement