1. Section 1.6
Functions

2. Relation, Domain, Range, and Function
Definitions
Relation, Domain, Range, and Function
The table describes a relationship between the variables x and y. This relationship is also described graphically.
x y
3 2
4 1
5 3 4
Section 1.6
Slide 2

3. Relation, Domain, Range, and Function
Definitions
Relation, Domain, Range, and Function
Definition
A relation is a set of ordered pairs.
The domain of a relation is the set of all values of the independent variable.
The range of the relation is the set of all values of the dependent variable.
Definition
Definition
Section 1.6
Slide 3

4. Relation, Domain, Range, and Function
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
Slide 4

5. 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
Slide 5

6. 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
Slide 6

7. Is the relation a function?
Deciding whether an Equation Describes a Function
Relation, Domain, Range, and Function
Example
Is the relation a function?
Solution
If x = 1, then
Input leads to two outputs: and
Therefore, the relation is not a function
Section 1.6
Slide 7

8. 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 is not a function
Section 1.6
Slide 8

9. Is the relation a function?
Deciding whether a Table is a Function
Relation, Domain, Range, and Function
Example
Is the relation a function?
x y
0 2
1 3
1 5 7
Solution
Input x = 1 leads to two outputs y = 3 and y = 5
So, the relation is not a function.
Section 1.6
Slide 9

10. Is the relation described by the graph a function?
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
Slide 10

11. Deciding whether a Graph Describes a Function
Vertical Line Test
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
Slide 11

12. Deciding whether a Graph Describes a Function
Vertical Line Test
Solution
All vertical lines intersects the curve at one point
Section 1.6
Slide 12

13. Deciding whether an Equation Describes a Function
Vertical Line Test
Example
Is the relation y = 2x + 1 a function?
Solution
Sketch the graph of
Each vertical line would intersect at just one point
So, the relation is a function
Section 1.6
Slide 13

14. 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
Slide 14

15. Properties of Linear Functions
The constant m is the slope of the line, a measure of the line’s steepness.
If m > 0, the graph of the function is an increasing line.
If m < 0, the graph of the function is a decreasing line.
If m = 0, the graph of the function is a horizontal line.
Section 1.6
Slide 15

16. Properties of Linear Functions
If an input increases by 1, then the corresponding output changes by the slope m.
If the run is 1, the rise is the slope m.
The y-intercept of the line is (0, b).
Since a linear equation of the form is a function, we know that each input leads to exactly out output.
Section 1.6
Slide 16

17. Rule of Four for Functions
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 a graph
3. a table, or words.
These four ways to describe input–output pairs of a function are known as the Rule of Four for functions.
Section 1.6
Slide 17

18. Describing a Function by Using the Rule of Four
Rule of Four for Functions
Example
Is the relation a function?
Since is of the form , it is a (linear) function.
List some input–output pairs of by using a table.
Solution
Example
Section 1.6
Slide 18

19. Describing a Function by Using the Rule of Four
Rule of Four for Functions
x y
– –2(–2) – 1 = 3
– –2(–1) – 1 = 1
–2(0) – 1 = –1
(1) – 1 = –3
–2(2) – 1 = –5
Solution
We list five input–output pairs of in the table on the right.
Describe the input–output pairs of using a graph.
Example
Section 1.6
Slide 19

20. Describing a Function by Using the Rule of Four
Rule of Four for Functions
Solution
We graph on the right.
Describe the input–output pairs of by using words.
Example
Solution
For each input–output pair, the output is 1 less than –2 times the input.
Section 1.6
Slide 20

21. 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.
Domain is the set of all x coordinates of the graph
Solution
No breaks in the graph
Leftmost point: (–4, 2),the rightmost point :(5, –3),
Domain is
Section 1.6
Slide 21

22. 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
Example
Using the graph of the function to determine the function’s domain and range.
Section 1.6
Slide 22

23. 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:
Section 1.6
Slide 23