1. Section 3.6
Functions

2. Understanding Relations
A relation is any set of ordered pairs. x y 1 2 3 4 1 2 3 4
y “depends” on the value we put in for x.
Independent variable x 3 3 y 4 2
Dependent variable

3. Example
State the domain and range of the relation. {(3, 5), (10, 12), (11, 8), (13, 15), (9, 12)} The domain consists of all the first coordinates in the ordered pairs. The range consists of all the second coordinates in the ordered pairs. The domain is {3, 9, 10, 11, 13}. The range is {5, 12, 8, 15}. We list 12 only once.

4. Function
A function is a relation in which no two different ordered pairs have the same first coordinate. x y 1 2 3 4 1 2 3 4
Domain x 3 3 y 4 2
Range

5. Example
State the domain and range of the relation and determine whether the relation is a function. {(2, 4), (5, 4), (3, 8), (2, 6), (1, 7)}
Domain
{(2, 4), (5, 4), (3, 8), (2, 6), (1, 7)}
Range
This is NOT a function because two different ordered pairs have the same first coordinate.
The domain is {2, 5, 3, 1}.
The range is {4, 8, 6, 7}.

6. Example Graph y = x2. Begin by constructing a table of values. x2
y = (2)2 = 4 4 1
y = (1)2 = 1 1
y = (0)2 = 0
y = (1)2 = 1 2
y = (2)2 = 4
Plot the ordered pairs and connect the points with a smooth curve.

7. Example Graph y = x2 + 3. Begin by constructing a table of values.2
(2)2 + 3 7 1
(1)2 + 3 4
(0)2 + 3 3 1
(1)2 + 3 2
(2)2 + 3
Plot the ordered pairs and connect the points with a smooth curve.

8. Example Graph x = y2 + 2. Begin by constructing a table of values.2
(2)2 + 2 6 1
(1)2 + 2 3
(0)2 + 2 2 1
(1)2 + 2
(2)2 + 2
Plot the ordered pairs and connect the points with a smooth curve.

9. Vertical Line Test
If a vertical line can intersect the graph of a relation more than once, the relation is not a function. If no such line can be drawn, then the relation is a function.

10. Example
Determine whether each of the following is the graph of a function. a. b. c. x y x y x y
Function
Not a Function
Not a Function

11. Example
If find each of the following. a. f(2) b. f(4) c. f(0)