1. 2.3 RELATIONS AND FUNCTIONS

2. CONSIDER y = 2x + 5
One way to display solutions is through a MAPPING DIAGRAM

3. Any set of ordered pairs is called a RELATION.
RELATIONS
Any set of ordered pairs is called a RELATION.
Ex: the set of points
{ (1, 0), (-2, 3), (4, -1), (9, 3) }
is a relation.

4. BUT…a line is a set of many points….
RELATIONS
We can use a graph to picture a relation.
BUT…a line is a set of many points….

5. …so, any equation y=mx+b is a relation.
RELATIONS
…so, any equation y=mx+b is a relation.

6. What is the domain and range for the set of points
The list of x-values used in a relation is called the DOMAIN.
The list of y-values used in a relation is called the RANGE.
What is the domain and range for the set of points
{ (1, 0), (-2, 3), (4, -1), (9, 3) }?

7. For { (1, 0), (-2, 3), (4, -1), (9, 3) } DOMAIN AND RANGE
We say the domain is D = { 1, -2, 4, 9 }.
We say the range is R = { 0, 3, -1 }

8. CONSIDER x = y2

9. FUNCTIONS
A FUNCTION is a relationship between two variables such that each value of the first variable is paired with exactly one value of the second variable.

10. CONSIDER x = y2
Therefore, x = y2 is NOT a function!

11. WHY??? FUNCTIONS { (1, 0), (-2, 3), (4, -1), (9, 3) }
{ (1, 0), (-2, 3), (1, -1), (9, 3) }
IS a function!!!
IS NOT a function!!!
WHY???

12. FUNCTIONS AND GRAPHS Is y = 2x + 1 a function?
List some ordered pairs…
( -1, -1 )
( 4, 9 )
( 1, 3 )
( 5, 11 )
( -2, -3 )
( 0, 1 )
( 3, 7 )
Could we ever re-use an x?

13. Any equation y = mx + b is a function.
FUNCTIONS AND GRAPHS
Since no x is repeated, y = 2x +1 IS a function!!!
Any equation y = mx + b is a function.

14. FUNCTIONS AND GRAPHS Does this graph represent a function?
HINT: What are coordinates of the marked points?

15. FUNCTIONS AND GRAPHS The circle is NOT a function!
The marked points are ( 0, 2 ) and ( 0, -2 )
The circle is NOT a function!

16. FUNCTIONS AND GRAPHS
Look at these graphs. Can you tell which are functions?

17. FUNCTIONS AND GRAPHS
The vertical line test:
If any vertical line intersects a graph at more than one point, the graph is NOT a function.

18. FUNCTIONS AND GRAPHS
This graph fails the VLT, and is NOT a function!

19. FUNCTIONS AND GRAPHS
This graph passes the VLT, and IS a function!

20. FUNCTIONS AND GRAPHS
This graph fails the VLT, and is NOT a function!

21. FUNCTION NOTATION
The height of an object x seconds after launch can be expressed by the function:
HOW TO SAY IT….”f of x”
x is the INDEPENDENT variable (input)
f(x) is the DEPENDENT variable (output)

22. FUNCTION NOTATION
What is the height of the object 3 seconds after launch?
What is the height of the object 4.5 seconds after launch?

23. THE END