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Section 5.2
Functions
Copyright © 2011 Pearson Education, Inc.
Relation, Domain, and Range
Definition
A relation is a set of ordered pairs. The domain of
a relation is the set of all values of the independent
variable, and the range of the relation is the set of
all values of the dependent variable.
We can think of a relation as a machine in which
values of x are “inputs” and values of y are “outputs.”
In general, each member of the domain is an input,
and each member of the range is an output.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 2
Relation, Domain, and Range
A relation described by
a table.
A relation described by
a graph.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 3
Relation, Domain, and Range
Think of a relation as an input-ouput machine.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 4
Function
Definition
A function is a relation in which each input
leads to exactly one output.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 5
Deciding whether an Equation
Describes a Function
Example 1
Is the relation y = x + 2 a function? Find the domain
and range of the relation.
Solution
Let’s consider some input-output pairs in the table
on the next slide.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 6
Deciding whether an Equation
Describes a Function
Solution continued
Each input leads to just one output-namely, the input
increased
by 2-so the
relation
y = x + 2 is
a function.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 7
Deciding whether an Equation
Describes a Function
Solution continued
The domain of the relation y = x + 2 is the set of
all real numbers, since we can add 2 to any real
number. The range of y = x + 2 is also the set of
real numbers, since any real number is the output
of the number that is 2 units less than it.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 8
Deciding whether an Equation
Describes a Function
Example 2
Is the relation y = ±x a function?
Solution
If x = 1, then y = ±1. So, the input x = 1 leads to
two outputs: y = −1 and y = 1. Therefore, the
relation y = ±x is not a function.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 9
Deciding whether an Equation
Describes a Function
Example 3
Is the relation y2 = x a function?
Solution
Let’s consider the input x = 4.
The input x = 4 leads to two outputs: y = −2 and
y = 2. So, the relation y2 = x is not a function.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 10
Deciding whether a Table
Describes a Function
Example 4
Is the relation described
by the table a function?
Solution
The input x = 1 leads to two outputs: y = 3 and
y = 5. So, the relation is not a function.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 11
Vertical-Line Test
A relation is a function if and only if every
vertical line intersects the graph of the relation at
no more than one point. We call this requirement
the vertical-line test.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 12
Deciding whether a Graph
Describes a Function
Example 6
Determine whether the graph represents a function.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 13
Deciding whether a Graph
Describes a Function
Solution
1. Since the vertical
line (red) sketched
intersects the circle
more than once, the
relation is not a
function.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 14
Deciding whether a Graph
Describes a Function
Solution
2. Each vertical line
sketched intersects
the curve at one
point. In fact, any
vertical line would
intersect this curve
at just one point. So,
the relation is a
function.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 15
Finding the Domain and Range
Example 9
Use the graph of the function to determine the
function’s domain and range.
1.
2.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 16
Finding the Domain and Range
Solution
The domain is the set of all x-coordinates of points in
the graph. Since there are no breaks in the graph, and
since the leftmost point is (−4, 2) and the rightmost
point is (5,−3), the domain is
−4≤ x ≤ 5. The range is the set
of all y-coordinates of points in
the graph. Since the lowest
point is (5, −3) and the highest
point is (2, 4), the range is −3 ≤
y ≤ 4.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 17
Finding the Domain and Range
Solution continued
The graph extends to the left and right indefinitely
without breaks, so every real number is an xcoordinate of some point in the graph. The domain is
the set of all real numbers.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 18
Finding the Domain and Range
Solution continued
The output −3 is the smallest number in the range,
because (1, −3) is the lowest point in the graph. The
graph also extends upward indefinitely without
breaks, so every number larger
than −3 is also in the range.
The range is y ≥ −3.
Copyright © 2011 Pearson Education, Inc.
Section 5.2
Lehmann, Elementary and Intermediate Algebra, 1ed
Slide 19