Functions vs. Relations A relation is a set of items that relates an input to an output. A relation can be written as a set of order pairs, by using an input-output table, or by mapping diagram As an input-output table Input Output 2 5 4 7 -1 15 0 0
As ordered pairs (Input, Output) (2,5) ( 4,7) (-1, 15) (0, 0) As a mapping diagram 2 4 1 0
5 7 15 0
A relation does not have to be only numbers. Study the following (Nic, Hockey) (Hector, Lacrosse) (Sari, Cheerleading) (Carol, softball) (Pete, Hockey) As a mapping diagram Nic
Hockey
Hector
Lacrosse
Sari
Cheer..
Carol Pete
Softball
Practice A relation is given below. Rewrite the relation using the other two notation methods. 1) (0,2), (1,4), (2,6), (3,8)
2. Input
9
Output 3
3.
9
25
25
-3
5
-5
足8 9 2
10 2 11 足4
A function is a special relation where for each input there is exactly one output. In a function you can say that the output is a function of the input. Study the following mapping diagram to see how a function and a relation are the different. Pay close attention to the lines being drawn from the input table. Function
Relation
3
2
3
2
5
4
5
4
7
6
7
6
9
8 10
9
Functions can be written using the same 3 notations as relations- ordered pairs, tables, and diagrams. (Equations and graphs are a 4th and 5th notation that can also be used for functions and we will learn more about these later.) As you can see one of the easiest ways to determine if a function is a relation is to use the mapping diagram to make sure that each input only goes to one output. Practice: Determine if each notation below represents only a relation or a function. 1. (-2,4), (2,4), (4,2), (-2,-4) 2. (3,-2), (6,1), (-3,5), (4,1) 3. Input -4 Output 8
0 0
4 8
8 32
4.
1 2 3 4 5
2 3 5 7 8
Homework Identify each relation as a function or just a relation 1. (4,5), (2,-3), (4,9), (-2,-3) 2. (-3,7), (3,7), (7,3) (-7,-3)
3. Input Output
2 3
1 -6
5 3
-1 -6
-8 10
9 2
2 11
-8 -4
4. Input Output
5.
13 14 16 22
30 35 12