1. Relations and Functions
2-1
Relations and Functions

2. Today’s objectives: Analyze and graph relations.
Find functional values.

3. Ordered Pairs
Ordered pairs are names of points in a coordinate plane. They are written in the form (x,y).
They are used to describe how the set of the x-coordinates (the domain) can be related to the set of the y-coordinates (the range).
Ordered pairs also determine the graph that represents a particular equation in the coordinate plane.

4. Cartesian Coordinate Plane
y - axis
Quadrant II
Quadrant I
origin (0,0)
x - axis
Quadrant III
Quadrant IV

5. Relation Def: a set of ordered pairs
The DOMAIN of a relation is the set of all x-coordinates from the ordered pairs, and the RANGE in the set of all y-coordinates.
The graph of the relation is the set of points in the coordinate plane corresponding to the ordered pairs in the relation.

6. Function
Def: a special type of relation in which each element of the domain is paired with exactly one element of the range.
We can use a mapping to show how each member of the domain is paired with each member of the range.
Ex: {(-3,1), (0,2), (2,4)} Domain Range

7. More Mapping Examples {(-3,1), (0,2), (2,4)} {(-1,5), (1,3), (4,5)}
{(5,6), (-3,0), (1,1), (-3,6)} m

8. More Mapping Examples {(-3,1), (0,2), (2,4)} one-to-one function
{(-1,5), (1,3), (4,5)} function; not one-to-one
{(5,6), (-3,0), (1,1), (-3,6)} not a function m

9. Example 1:
State the domain and range of the relation shown in the graph. Is the relation a function?
(3,3)
(-3,1)
(1,2)
(-4,0)
(0,-2)

10. Example 2:
Why is {(9,3), (9,-3), (4,2), (4, -2)} not a function?

11. Vertical Line Test
The vertical line test can be used to determine whether or not a relation is a function by simply examining its graph.
Ex:

12. Vertical Line Test
The vertical line test can be used to determine whether or not a relation is a function by simply examining its graph.
Ex:
function
not a function

13. Example 3:
Graph the following data, and then tell whether or not the relation is a function.
YEAR
FUEL EFFICIENCY
1995
20.5 mi/gal
1996
20.8 mi/gal
1997
20.6 mi/gal
1998
20.9 mi/gal
1999
2000
2001
20.4 mi/gal

14. Equations and Relations
Relations and functions can also be represented by equations.
To find values for the relation, we make a table.
Ex: For y = 2x + 1… X Y -1 1 2

15. Example 4: Graph the relation represented by y = 3x – 1.
Find the domain and range.
Determine whether or not the relation is a function.

16. Example 5: Graph the relation represented by x = y2 + 1.
Find the domain and range.
Determine whether or not the relation is a function.

17. Functions (continued)
When an equation represents a function, the variable x (the domain) is called the INDEPENDENT VARIABLE, and y is the DEPENDENT VARIABLE because its value depends on x.
Because the values of x determine the values of y, we can use FUNCTIONAL NOTATION, which more or less replaces y with f(x).
Ex. y = 2x + 1 can be written as f(x) = 2x + 1.
Using this notation, I can evaluate the function for a particular value of x.

18. Example 6:
Given f(x) = x3 -3 and h(x) = 0.3x2 – 3x – 2.7, find each value.
a) f(-2)
b) h(1.6)
c) f(2t)

19. HOMEWORK
p. 60 #17-22, even, 35-41