1. 2-1: Relations and Functions
Essential Question: What separates a relation from a function?
2-1: Relations and Functions

2. 2-1: Relations and Functions
A relation is a set of pairs of input and output values. You can write a relation as a set of ordered pairs.
Input (time): { }
relation: {(0, 10), (0.1, 9.8), (0.2, 9.4), (0.3, 8.6), (0.4, 7.4)}
Output (height): 0, {10 0.1, , , , 7.4}
The 1st input goes with the 1st output, 2nd input with 2nd output, 3rd input with 3rd output, etc.

3. 2-1: Relations and Functions
You can graph a relation on a coordinate plane.
Example 1: Graph the relation { , , , }
The first number represents the “x” (left/right)
The second number represents the “y” (up/down)
(-2, 4)
(3, -2)
(-1, 0)
(1, 5)

4. 2-1: Relations and Functions
Your Turn
Graph the relation: {(0, 4), (-2, 3), (-1, 3), (-2, 2), (1,-3)}

5. 2-1: Relations and Functions
The domain of a relation is the set of all inputs (x-coordinates)
The range of a relation is the set of all outputs (y-coordinates)
The way to keep all that straight?
“d” comes before “r”
“x” comes before “y”
So, in an ordered pair, the first number is the domain & x-coordinate. The second number is the range & y-coordinate

6. 2-1: Relations and Functions
Example 2: Finding domain and range from a graph.
Find the domain and range from the relation below.
Relation:
{(-3, 2), (0, 1), (2, 4), (4, -3)}
Domain:
{-3, 0, 2, 4}
Range:
{-3, 1, 2, 4}

7. 2-1: Relations and Functions
Your Turn
Find the domain and range from the relation below.
Relation:
{(-3, 1), (-1, 1), (1, 1), (1, 3), (-1, -2), (-1, -4), (1, -4)}
Domain:
{-3, -1, 1}
Range:
{-4, -2, 1, 3}

8. 2-1: Relations and Functions
Another way to show a relation is by a mapping diagram. A mapping diagram places the domain and range in boxes, and draws arrows to connecting elements.
Example 3: Make a mapping diagram for the relation {(-1, -2), (3, 6), (-5, -10), (3, 2)}
Domain
Range
-10 -2 2 6 -5 -1 3

9. 2-1: Relations and Functions
Your Turn: Make a mapping diagram for the relation {(2, 8), (-1, 5), (0, 8), (-1, 3), (-2, 3)}
Domain
Range 3 5 8 -2 -1 2

10. 2-1: Relations and Functions
A function is a relation where each element of the domain is paired to exactly one element in the range
Meaning: No element of the domain gets repeated
Example 4: Using mapping diagrams
Domain
Range
Domain
Range -2 5 -1 3 4 -1 2 3 -1 3 5
Is a function
Not a function

11. 2-1: Relations and Functions
Your Turn: Which of the following are functions?
Domain
Range
Domain
Range 2 3 4 7 5 6 8 -1 1 -3 7 10
Is a function
Not a function

12. 2-1: Relations and Functions
If we’re given a graph, we can use the vertical line test to determine whether a relation is a function
We make a vertical (up/down) line with some straight object (ruler, pencil), and move it from left to right.
If the graph ever touches our line more than once, it is not a function
Not a function
Is a function

13. 2-1: Relations and Functions
Your Turn: Which of the following are functions?
Is a function
Not a function

14. 2-1: Relations and Functions
A function rule expresses an output value in terms of an input value.
y = 2x
f(x) = x + 5
C = πd
output input
You read function notation f(x) as “f of x” or “a function of x”.
Note that it doesn’t mean “f times x”
Whenever you get a value in the parenthesis, it means you substitute that value for x in the function.
Example: f(3) = = 8

15. 2-1: Relations and Functions
Find f(-3), f(0) and f(5) for each function.
Example 6: f(x) = 3x – 5
f(-3) = 3(-3) – 5 = -9 – 5 = -14
f(0) = 3(0) – 5 = 0 – 5 = -5
f(5) = 3(5) – 5 = 15 – 5 = 10
Your turn: f(a) = ¾a – 1
f(-3) =
f(0) =
f(5) =
¾(-3) – 1 = -9/4 – 1 = -13/4
¾(0) – 1 = 0 – 1 = -1
¾(5) – 1 = 15/4 – 1 = 11/4

16. 2-1: Relations and Functions
Assignment
Page 59
Problems 1 – 29 (odd problems)