1. Definition
Function:
A function is a relation that assigns exactly one value of the range to each of the values in the domain.
*Each x value must be associated with one, and only one y value.
We will look at a number of examples so that we can begin to get a feeling for what exactly is meant by the term “function”

2. Since for each x value there is one and only one y value, this is indeed a function.
Is the relation represented in this table of values a function? X Y 11 -2 12 -1 13 20 7
In order to answer this question, we have to see if for each x value, there exactly one y value associated with it.
(11,-2)
(12,-1)
(13,0)
(20,7)

3. The relation is not a function because two y values (-1 and 1) are assigned to one x value (-2)
Is this a function? X Y -2 -1 -3 6 3 1
(-2,-1)
(-3,0)
(6,3)
(-2,1)

4. 1 1 2 5 6 9 14 20 Draw a mapping for the relation below.
{(1,5), (2,9), (6,20), (14,1)}
Input Values
Output Values 1 2 6 14 1 5 9 20

5. Function or not a function? Explain.
Input
Output 1 2 6 14 1 5 9 20

6. Function or not a function? Explain.
Domain
Range 9
113 8 3 7 12 5

7. Function or not a Function? Explain.
Input
Output A B C D E F 1 2 3 4 5 6

8. Function or not a function? Explain.
Domain
Range 73 14 20 19 26

9. Function or not a function? Explain.
Range
Domain 8 35
100 9

10. Function or not a function? Explain.
Range
Domain
Paul
Mark
Ernie
Maria
January
February
March
April
May

11. Function or not a function? Explain.
Range
Domain
1000
2000
3000
4000 17 45 A 3

12. Function or not a function? Explain.
Range
Domain 4
1007 9 14 73 8 10 12 14 11

13. Function or not a function? Explain.
Output
Input
A-Rod
Jeter
Sheffield
Matsui 19 23 48 34

14. Function or not a function? Explain.
Range
Domain 37
921
742 12 10 27 10 20

15. Cowboys and Horses
The x values (domain) can be thought of as horses. (Since they have four legs like a horse.)
The y values (range) can be thought of as cowboys. (Since cowboys stand up like a “Y”)
A horse cannot be owned by more than one cowboy. (An x value cannot correspond to more than one y value).
A cowboy however can own more than one horse. (A y value can correspnond to more than on x value.)
There are no “wild horses”. (There is no x value that is not related to a y value.)

16. Determine whether or not each relation is a function
Determine whether or not each relation is a function. (If it helps you may want to draw a mapping)
1) {(2,7), (5,12), (-3, 14), (11, 20), (4,9)}
Function
2) {(4,12), (9,13), (2, 7), (4, 11), (5, 1)}
Not a Function
3) {(4,6), (6,4)}
Function
4) {(1,5), (-1,5), (2,5)}
Function
5) {(-3,2), (4,-3), (8,121), (5,18), (6,12)}
Function
6) {(1,7), (2,14), (3,21), (4, 28), (2, 35)}
Not a Function
Alternate Definition of a Function: A function is a relation in which no two ordered pairs have the same first element.

17. Function Rule: Definition y = 3x + 4
A function rule is an equation that describes a function. If you know the input values of a function, you can use a function rule to find the output values
If you plug in a value for the independent variable (x), you will get a value for the dependent variable (y).
Example:
y = 3x + 4
Input values
Independent Variable
Output values
Dependent Variable

18. It may help to view a function rule as a machine
It may help to view a function rule as a machine. You put an input value into the machine and it gives you an output value. X
Input
Function Machine
Output Y

19. (1,7)
(2,10) 3 2 1
(3,13)
Input
y = 3x + 4
Output 10 13 7

20. Plug each of the following input values into the function and list each input value with its associated output value as ordered pairs.
{-2,-1,0,1,2}
(-2,13) (-1,9) (0,5) (1,1) (2,-3)
Input
y = -4x + 5
Output
{13,9,5,1,-3}

21. Find the range of the following functions when the domain is {-4, -1, 0, 2, 8}
1) y = 4x + 2
2) p = -2m – 8
3) y = x² + 2

22. We have already learned how to make a table of values and use it to graph equations. There is a quick way to determine whether or not an equation is a function by looking at its graph.

23. On a coordinate grid, plot the following points.
{(1,4) (-2,6) (0,3) (-5,-3) (1,-1) (2,6)}
Is this relation a function?
How can someone tell, by just looking at the graph, whether or not the relation is a function?

24. Vertical-Line Test
If a vertical line passes through a graph more than once at any point as you slide it across the grid, then the graph is not a function. Y
Since it did not, the graph is a function.
Does this graph represent a function?
Did the vertical line cross the graph at two places at any time?
Use the vertical line test to answer the question. X

25. No. It is not a function since a vertical line hits the graph at more than one point.
Does this graph represent a function? y x

26. Why does the vertical line test works
Why does the vertical line test works. (Hint: how does the vertical line test relate to the definition of a function?).

27. Y X

28. Y X

29. Y X

30. Y X

31. Y X

32. Function Notation f(x) = 2x - 5
To write a rule in function notation, you use the symbol f(x) in place of y. You read f(x) as “f of x”.
To write the function y = 2x – 5 in function notation, you need to replace the dependent variable “y” with “f(x)”.
f(x) = 2x - 5
Dependent variable
Independent variable

33. In order to find f(-2), follow the following steps
Function notation allows you to see the input value. Suppose the input value from the previous equation was -2.
In order to find f(-2), follow the following steps
f(x) = 2x - 5
Rewrite the function rule
f(-2) = 2(-2) - 5
Everywhere the independent variable appears, replace it with the given input value
f(-2) = -9
Simplify
When the input is -2, the output is -9.

34. Error Alert
f(x) does not mean “f times x”.
Also
f is not the only letter that can be used for a function. g(x), h(x)…can also be used to represent a function

35. Given f(x) = 3x2 – 2, find each of the following:

36. f(-4) = 4 f(-3) = 5 f(-2) = -1 f(0) = 3 f(5) = 2
Given the graph below, find each of the following:
f(-4) = 4
f(-3) = 5
f(-2) = -1
f(0) = 3
f(5) = 2