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Section 1.6
Functions
Definitions
Relation, Domain, Range, and Function
The table describes a relationship between the
variables x and y. This relationship is also described
graphically.
Section 1.6
x
y
3
4
5
5
2
1
3
4
Lehmann, Intermediate Algebra, 3ed
Slide 2
Definitions
Relation, Domain, Range, and Function
Definition
A relation is a set of ordered pairs.
Definition
The domain of a relation is the set of all values of
the independent variable.
Definition
The range of the relation is the set of all values of
the dependent variable.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 3
Definitions
Relation, Domain, Range, and Function
Think of a relation as a machine where:
• x are the “inputs”
-Each member of the domain is an input.
• y are the “outputs”
-Each member of the range is an output.
Definition
A function is a relation in which each input leads to
exactly one output.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 4
Deciding whether an Equation Describes a Function
Relation, Domain, Range, and Function
Example
Is the relation y = x +2 a function? Find the domain
and the range of the relation.
Solution
Consider some
input-output pairs.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 5
Deciding whether an Equation Describes a Function
Relation, Domain, Range, and 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.
Domain
• We can add 2 to any real number.
• So, the domain is the set off all real numbers.
Range
• Output is two more than the input.
• So, the range is the set off all real numbers.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 6
Deciding whether an Equation Describes a Function
Relation, Domain, Range, and Function
Example
Is the relation y   x a function?
Solution
• If x = 1, then y  1
• Input x  1 leads to two outputs: y  1 and y  1
• Therefore, the relation y   x is not a function
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 7
Deciding whether an Equation Describes a Function
Relation, Domain, Range, and Function
Example
Is the table a function?
Solution
• Consider the input x = 4
• Substitute 4 for x and solve for y:

• Input x = 4 leads to two outputs: y = –2 and y = 2
• So, the relation y 2  x is not a function
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 8
Deciding whether a Table is a Function
Relation, Domain, Range, and Function
Example
Is the relation y  x a function?
2
Solution
• Input x = 1 leads to two
outputs y = 3 and y = 5
• So, the relation is not a
function.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
x
0
1
1
2
3
y
2
3
5
7
10
Slide 9
Definition
Relation, Domain, Range, and Function
Example
Is the relation described by
the graph a function?
Solution
• The input x = 1 leads to two
outputs: y = –4 and y = 4
• So, the relation is not a
function
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 10
Deciding whether a Graph Describes a Function
Ve r t i c a l L i n e T e s t
Solution
• The vertical line sketched
intersects the circle more
than once
• The relation is not a
function.
Example
Is the relation described by the graph on in the next
slide a function?
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 11
Deciding whether a Graph Describes a Function
Ve r t i c a l L i n e T e s t
Solution
• All vertical lines intersects the curve at one point
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 12
Deciding whether an Equation Describes a Function
Ve r t i c a l L i n e T e s t
Example
Is the relation y = 2x + 1 a function?
Solution
• Sketch the graph of
y  2x  1
• Each vertical line would
intersect at just one point
• So, the relation is a function
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 13
Definition and Properties
Linear Function
Definition
A linear function is a relation whose equation can
be put into the form
y = mx + b
where m and b are constants.
Properties
Properties of linear functions:
1. The graph of the function is a nonvertical line.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 14
Properties of Linear Functions
Linear Function
2. The constant m is the slope of the line, a measure
of the line’s steepness.
3. If m > 0, the graph of the function is an
increasing line.
4. If m < 0, the graph of the function is a decreasing
line.
5. If m = 0, the graph of the function is a horizontal
line.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 15
Properties of Linear Functions
Linear Function
6. If an input increases by 1, then the corresponding
output changes by the slope m.
7. If the run is 1, the rise is the slope m.
8. The y-intercept of the line is (0, b).
Since a linear equation of the form y  mx  b is a
function, we know that each input leads to exactly
out output.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 16
Definition
Rule of Four for Functions
Definition
We can describe some or all of the input–output
pairs of a function by means of
1. an equation
2. a graph
3. a table, or
4. words.
These four ways to describe input–output pairs of a
function are known as the Rule of Four for
functions.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 17
Describing a Function by Using the Rule of Four
Rule of Four for Functions
Example
Is the relation y  2 x  1 a function?
Solution
Since y  2 x  1 is of the form y  mx  b, it is a
(linear) function.
Example
List some input–output pairs of y  2 x  1 by using
a table.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 18
Describing a Function by Using the Rule of Four
Rule of Four for Functions
x
Solution
We list five input–output
pairs of y  2 x  1 in the
table on the right.
Example
Describe the input–output
pairs of y  2 x  1 using a
graph.
Section 1.6
–2
–1
0
1
2
Lehmann, Intermediate Algebra, 3ed
y
–2(–2) – 1 = 3
–2(–1) – 1 = 1
–2(0) – 1 = –1
2(1) – 1 = –3
–2(2) – 1 = –5
Slide 19
Describing a Function by Using the Rule of Four
Rule of Four for Functions
Solution
We graph y  2 x  1 on
the right.
Example
Describe the input–output
pairs of y  2 x  1 by
using words.
Solution
For each input–output pair, the output is 1 less than
–2 times the input.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 20
Finding the Domain and Range
Using a Graph to Find the Domain and Range of a Function
Example
Using the graph of the
function to determine the
function’s domain and range.
Solution
• Domain is the set of all x
coordinates of the graph
• No breaks in the graph
• Leftmost point: (–4, 2),the rightmost point :(5, –3),
• Domain is 4  x  5.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 21
Finding the Domain and Range
Using a Graph to Find the Domain and Range of a Function
Solution Continued
• Range is the set of all y-coordinates of points
• Lowest point is (5, –3), highest is (2, 4)
• The range is 3  y  4
Example
Using the graph of the
function to determine the
function’s domain and range.
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 22
Finding the Domain and Range
Using a Graph to Find the Domain and Range of a Function
Solution
Domain
• Extends left and right indefinitely without breaks
• Domain: set of all real numbers
Range
• Lowest point is (1, –3)
• Highest is indefinite without breaks
• Range: y  3
Section 1.6
Lehmann, Intermediate Algebra, 3ed
Slide 23