Chapter 4 Introduction to Relations and
Functions For Exercises 1124, find the domain
and range of the relations. Use interval notation
where appropriate. 11. 12. 13. 14. 15. 16. 17.
18. 19. 20. y x 5 1 4 3 2 3 4 5 2 1 3 4 5 1 2 1 3 4
52yx5143234521345121
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 mor
can be found using a Table feature. Function
values can also be evaluated by using a Value
(or Eval) feature. The value of g142 is shown
here. Y1 1 2x 1 g1x2 Calculator Connections
Skill Practice Answers 6a. b. c. 3 d. 4 5 3 Section
4.2 Introduction to Fu
Skills 29. a. Define a relation with four ordered
pairs such that the first element of the ordered
pair is the name of a friend and the second
element is your friends place of birth. b. State
the domain and range of this relation. 30. a.
Define a relation
Set of second coordinates 1. Find the domain
and range of the relation. e 10, 02, 18, 42, a 1
2 , 1b, 13, 42, 18, 02 f Skill Practice Example 1
Table 4-1 Definition of a Relation in x and y Any
set of ordered pairs (x,y) is called a relation in x
and y. F
Relation 2. Applications Involving Relations
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Page 256 IA CONFIRMING PAGES Section 4.1
Introduction to Relations 257 Finding the
Domain and Range of a Relation Find the
domain and range of the relation cfw_(Alabama, 7
Definition of a Quadratic Function 4. Finding the
x- and yIntercepts of a Function Defined by y = f
(x ) 5. Determining Intervals of Increasing,
Decreasing, or Constant Behavior Definition of a
Linear Function and a Constant Function Let m
and b represent
2. The shape of the graph of a quadratic
function. 3. A function whose graph is a
horizontal line. 6. A function whose graph is a
line that is not vertical or horizontal. 7. The set
of second coordinates of a set of ordered pairs.
1 4 2 3 5 6 7 IA 4 miL28
the function at . Use these values to support
your answer to Exercise 80. 99. a. Graph on a
viewing window defined by and b. Use the
graph to approximate the function at . Use
these values to support your answer to Exercise
79. t 1 h1t2 16t 0 t 2 0 y 100.
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 keep these
guidelines in mind. Exclude values of x that
make the denominator of a fr
more than once y x Not a Function A vertical
line intersects in more than one point. 5 1 4 3 3
4 5 2 1 3 4 5 1 2 1 4 5 2 3 Function No vertical
line intersects more than once. y x 5 1 4 3 3 4 5
2 1 3 4 5 1 2 1 4 5 2 3 2 511, 42, 12, 12, 13, 22 6
511, 32,
not f x. miL2872X_ch04_255-308 9/25/06 11:52
AM Page 268 CONFIRMING PAGES d. We say g
of is . This is equivalent to the ordered pair
Notice that and correspond to the ordered pairs
and In the graph, these points line up. The
graph of all ordered pairs def
be defined by several different methods: by a
list of ordered pairs, by a correspondence
between the domain and range, by a graph, or
by an equation. Skill Practice Example 2 The xand y-components that constitute the ordered
pairs in a relation do not nee
Introduction to Relations and Functions 4.1
Introduction to Relations 4.2 Introduction to
Functions 4.3 Graphs of Functions 4.4 Variation
255 In this chapter we introduce the concept of
a function. In general terms, a function defines
how one variable dep
2 x2 9 x2 Section 4.2 Introduction to Functions
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PM Page 271 IA CONFIRMING PAGES 272
Chapter 4 Introduction to Relations and
Functions Study Skills Exercise 1. Define the key
terms. a. Function b. Function notation c
Section 4.1 Introduction to Relations 263 21.
22. 23. 24. Concept 2: Applications Involving
Relations 25. The table gives a relation between
the month of the year and the average
precipitation for that month for Miami, Florida.
a. What is the range elemen
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PAGES Section 4.1 Introduction to Relations 261
Study Skills Exercises 1. Compute your grade at
this point. Are you earning the grade that you
want? If not, maybe organizing a study group
would help. In a study gro
U.S. National Oceanic and Atmospheric
Administration Age Maximum Recommended
Heart (years) Rate (Beats per Minute) y x 20 200
30 190 40 180 50 170 60 160
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Page 263 IA CONFIRMING PAGES 264 Chapter 4
Introduction to Rel
as a set of ordered pairs. 3. 4. 5. 6. 7. List the
domain and range of Exercise 3. 8. List the
domain and range of Exercise 4. 9. List the
domain and range of Exercise 5. 10. List the
domain and range of Exercise 6. C 3 D 4 A 1 B 2
E 5 x y Year of State,
height. Based on these data, a forensics
specialist or archeologist can find a linear
relationship between height y and femur length
x: From this type of relationship, the height of a
woman can be inferred based on skeletal
remains. a. Find the height of
and constant functions, the following equations
define six basic functions that will be
encountered in the study of algebra: Equation
Function Notation The graph of the function
defined by is linear, with slope and y-intercept
(0, 0) (Figure 4-9). To dete
Find the domain and range of the relations:
Solution: a. Domain: cfw_3, 2, 7 Range: cfw_9
Example 3 Figure 4-3 y x 5 4 3 2 1 1 2 3 4 5 5 4 3
2 1 1 2345 x y2 2 3 7 9 x y Figure 4-1 3 1 3 4 2 x
y Domain Range 4 A relation may be defined
as a set of ordered
11); domain: cfw_Bear, Cat, Deer, Dog, range:
cfw_22.5, 11, 12.5 IA Animal, Longevity (years), x y
Bear 22.5 Cat 11 Deer 12.5 Dog 11
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Page 257 CONFIRMING PAGES 258 Chapter 4
Introduction to Relations and Functions A
r
42 5 g1x2 3x 5 t 2 2t f1t2 1t2 x t 2 21t2 f1x2 x2
2x f1t2 g1w 42 f1t2 f1x2 x g1x2 3x 5, 2 2x
Example 4 x 5 4 3 2 1 123 4 5 5 y h(x) 1 4 3 2 2 1
3 4 5 h(x) Figure 4-6 Substitute for all values of x
in the function. Simplify. x w 4 Skill Practice
Answers 7a
domain because any real number substituted
for t will produce a real number. The domain is
the set of all real numbers. Interval notation:
Write the domain of the functions in interval
notation. 12. 13. 14. g1x2 1x 2 15. h1x2 x 6
p1x2 5 4x2 1 f1x2 2x 1 x
element in the domain? d. Complete the
ordered pair: (50, ) e. Complete the ordered
pair: ( , 190) 27. The population of Canada, y, (in
millions) can be approximated by the relation
where x represents the number of years since
2000. a. Approximate the pop
and the range elements are the y-coordinates.
Domain: cfw_2, 1, 0, 1, 2 Range: cfw_3, 0, 1 c. The
domain consists of an infinite number of xvalues extending from 8 to 8 (shown in red).
The range consists of all y-values from 5 to 5
(shown in blue). Thus,
and 8 x 86 or 38, 84 5x x is a real number
and y x Skill Practice Answers 3. Domain cfw_5, 2,
4, range cfw_0, 8, 15, 16 4. Domain cfw_4, 0, 1, 4,
range cfw_5, 3, 1, 2, 4 5. Domain: or [4, 0], range:
or [2, 2] 6. Domain: , range: 1, 04 1, 2 2 y 26 5y
y is
graph in more than one point, the relation
cannot be a function. This idea is stated
formally as the vertical line test. Determine if
the relations define y as a function of x. 1. 2.
cfw_( ), ( ), ( ), ( ) 3. cfw_( ), ( ), ( ), ( ) 1, 6 8, 9 1, 4 3, 10
4,
y in the range. Section 4.2 Introduction to
Functions Concepts 1. Definition of a Function
2. Vertical Line Test 3. Function Notation 4.
Finding Function Values from a Graph 5.
Domain of a Function To understand the
difference between a relation that is a
horizontal line, as shown in Figure 4-7. We say
that a function defined by is a constant function
because for any value of x, the function value is
constant. An equation of the form is
represented graphically by a line with slope m
and y-intercept (0, b).
notation: a, 1 2 b a 1 2 , b x 1 x 2 1 2 2x 1 2x 1
0 g1t2 t k1t2 1t 4 2 3t h1x2 x 4 x2 9 f1x2 x 7 2x
1 Example 6 b. The quantity is greater than or
equal to 0 for all real numbers x, and the
number 9 is positive. Therefore, the sum must
be positive for al
defined by , find the function values. a. b. c. 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 ycoordinate of a point with x-coordinate a.
Finding Function Values