Identifying Relations and FunctionsA relation is a set of ordered pairs.•The domain of the relation is x-coordinate of the ordered pair. It is also considered as the input (independent variable).

•The range of the relation is y-coordinate of the ordered pair. It is also considered as the output (dependent variable).

DomainRange

x 1 2 3 4

y 5 7 9 11

InputOutput

Another way to understand it is…

Understanding Functional

Question: How do I buy some M&M’s without breaking the vending machine?

Correct Answer: After I put my money in, I need to INPUT a value in order to get the M&M’s. But what’s missing?

Correct Answer: The numbers! Let’s add some in.

1 2

3 4

5 6

Understanding FunctionalIn order for something to be functional, you should know

EXACTLY what given to you (the output) after you input your choice.

Question: This is functional?

1 2

3 4

5 6

Input(Domain) Output(Range)1 granola bar2 pretzels3 popcorn4 chips5 M&M’s

Yes, this is functional

Understanding FunctionalLet’s change the input around

Question: This is functional? Do I know exactly what I will get after I input a choice?

1 2

3

6

Input(Domain) Output(Range)1 granola bar2 pretzels3 popcorn

chipsM&M’sNo, this is not

functional

4

5

3

3

Understanding FunctionalAgain, let’s make changes

Question: This is functional? Do I know exactly what I will get after I input a choice?

1 2

3 4

5 6

Input(Domain) Output(Range)1 granola bar23 popcorn

M&M’s456

Yes, this is functional

Is this a function?

For a relation to be a function, one input (x) must have exactly one output (y).

Domain Range

01

12

234

For example, is this a function? Explain.

This is NOT a function; the input of 1 has two different outputs.

Domain Range0

11

223 4

This is a function; all inputs have exactly one output.

Examples

Domain Range0 11 32 2

4

Is this a function? Explain.

This is a function, all inputs have exactly one output.

A) (0,1), (1,3), (2,2), (3,4)

3

Mapping a diagram can be helpful

Domain Range-2 32 2

-2This is NOT a function; the input of 2 has two different outputs.

B) (-2,3), (2,2), (2,-2)

Graphing Relations and FunctionsLet’s graph Example B to see how it looks.

Remember, this graph shows

something NOT functional.

B) (-2,3), (2,2), (2,-2)

x

y

1 2 3 4

Let’s see another graph NOT functional.

x

y

1 2 3 4

Question: Why is this NOT functional?

Graphing Relations and FunctionsVertical Line Test

x

y

1 2 3 4

x

y

1 2 3 4

If you can find a vertical line that passes through two points on the graph, then the relation is NOT a function.

Use your pencil as a vertical line, and check.Oh no!!!

TWO points!

TWO points! Failed!

Function Rule

• Describes the operation performed on the domain to get the range.

• When written as an equation it is a function notation.

Function Notation f(x)

Equation

• y= 2x +3• Solve for y if x=4• Y=2(4) +3• Y=8+3• Y=11

Function Notation

( ) 2 3f x x

(4) 2(4) 3

(4) 8 3

(4) 11

f

f

f

f(x)=6x+5, find each function value

a. f(7)

f(7) = 6(7) +5f(7)=42 +5f(7)=47

b. f(-4)f(-4) = 6(-4) +5f(-4) = -24 +5f(-4)= -19

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