1. Introduction to Functions
Unit 4
Lesson 1

2. Objective: Define terms related to functions
Determine if a relation is a function
Evaluate a function

3. Definitions
An ordered pair – used to locate points on a graph. (x, y) The first number, the x value corresponds to the horizontal axis The second number, the y value, corresponds to the vertical axis

4. Definition
Relation – a set of ordered pairs

5. 5 Ways to Represent a Relation
Set of Ordered Pairs
Mapping
Table
Graph
Equation or function

6. Example
Represent the following relation 4 ways: {(-2, -9), (-1, -5), (0, -1), (1, 3), (2, 7)}
Mapping:
X values
Y values -2 -1 1 2 -9 -5 -1 3 7

7. {(-2, -9), (-1, -5), (0, -1), (1, 3), (2, 7)} Table: Graph: X Y -2 -91 3 2 7
Equation:

8. Definitions
Domain – input values or x- values of a relation
Range – output values or the y-values of a relation

9. Example {-4, -3, 0, 2, 3} Find the Range: {-4, -1, 0, 1, 2}
Given the relation: {(-3, 2), (0,-4), (2,0), (-4, -1), (3, 1)} Find the domain:
{-4, -3, 0, 2, 3}
Find the Range:
{-4, -1, 0, 1, 2}

10. Given the graph Find the domain: Find the range:
{-4, -2, 2, 3}
{-5, -4, 1, 5}

11. Definition
A function is a relation such that for every x there is one and only one y (X can’t repeat)

12. Examples
Determine if the following are functions. If not a function, explain why.
{(0,2), (1,5), (6,3), (7,-5)}
{(-3, 1), (-2, 6), (-1,0), (1, 0), (2, 6)}
{(0, 2), (4,3), (0, -2)}
Function
Function
Not a Function

13. Example Is this a function? (Not in notes)y 2 1 -1 2
Yes

14. Is this a function? x y -3 -1 4 -1 3 5
Yes

15. Is this a function? x y -7 -9 8 2 2 5 7
Yes

16. Is this a function? -3 -2 4 -5 8 10 12 No

17. Is this a function? -1 -2 -3 -4 3 4 5 No

18. Is this a function? X Y -2 -1 7 2 3
Yes

19. Is this a function X Y -2 -1 1 2 7 No

20. Another Definition
The vertical line test – If a vertical line intersects the graph more than once, then the graph is not a function

21. Determine if the following are functions and give the range & domain of the each
Yes
(-∞, 5]
(-∞, 2]

22. Determine if the following are functions and give the range & domain of the each
Yes
(-4, ∞)
(-1, ∞)

23. Determine if the following are functions and give the range & domain of the each
(-∞, 3]
[-2, ∞)

24. Determine if the following are functions and give the range & domain of the each
Yes
(-∞, ∞)
(-∞, ∞)

25. Determine if the following are functions and give the range & domain of the each
[0, ∞)
(-∞, ∞)

26. Determine if the following are functions and give the range & domain of the each
Yes
[-6, 6]
[0, 6]

27. Determine if the following are functions and give the range & domain of the each
[-4,4]
[-4,4]

28. Determine if the following are functions and give the range & domain of the each
[-5]
(-2, 6)

29. Function Notation
Read as f of x
Does not mean multiply f times x

30. Evaluating a function
To evaluate a function, replace the ‘x’ with the ‘number’ in the function, and then do the math.

31. Examples Find f(3) Every place there is an x, replace it with 3

32. Examples Find f(-1) Every place there is an x, replace it with -1

33. Examples Find f(a) Every place there is an x, replace it with a
f(a) = 2(a) - 4
f(a) = 2a - 4

34. Examples Find f(a-3) Every place there is an x, replace it with a - 3
f(a=3) = 2(a-3) - 4
f(a - 3) = 2a - 6 – 4
f(a – 3) = 2a - 10

35. Example 2
Given: f(x) =2x2 + x -3 Evaluate f(-1) f(-1) = 2(-1)2 + (-1) -3 f(-1) = – 3 = -2

36. Example 2
Given: f(x) =2x2 + x -3 Evaluate f(-3) f(-1) = 2(-3)2 + (-3) -3 f(-1) = 18 + (-3) – 3 = 12

37. Example 3 Replace the x with ‘a’ f(a) = f(x) =2a2 + a -3
Given: f(x) =2x2 + x -3 Find f(a)
Replace the x with ‘a’
f(a) = f(x) =2a2 + a -3

38. Example 3 Replace the x with ‘a + 1’ f(a+1) = f(x) =2(a+1)2 + (a+1) -3
Given: f(x) =2x2 + x -3 Find f(a+1)
Replace the x with ‘a + 1’
f(a+1) = f(x) =2(a+1)2 + (a+1) -3
Simplify. Remember to FOIL!

39. Quick Review
What is a relation? What is a domain? What is the range? What is a function How do you say this: f(x) What is the vertical line test