Algebra

4.8 Functions and Relations

Functions and Relations A relation is a set of ordered pairs A function is a rule that establishes

a relationship between two quantities, called the input and the output.

Input = x The collection of the x values is known as the domain.

Output = y. The collection of the y values is known as the range.

Function

The most important part of the function: every input has only one output. An x cannot have two y’s.

Examples:x y

1

2

3

4

2

4

4

5

Is this relation a function?

Yes. Each input has only one output.

x y

1

1

2

3

5

7

8

9

Is this relation a function?

No. The input 1 has Two outputs 5 and 7.

A Birthday Example

Imagine this relation: Input = All the students at CHSFA Output = The Birthday of that

Student Is this a function?

Let’s See

Person Birthday

Jessica Jones May 5th

Bob Boone June 21st

Sally Smith Nov. 18th

Todd Thomas Nov. 18th

Is this a function?

Does each input have only one output?

Similarly, does each person have only one birthday?

Since each input has only one output, then this is a function.Todd Thomas April 3rd

This would make this not a function because an input (Todd Thomas) would have more than one Output (Nov. 18th and April 3rd). This cannot happen.

What if the input were the days of the year and the output

was the people who had that day as their birthday? Let’s try as a class.

DatePeople’sB-Day

Jan. 1st Stu Sanders

Jan. 2nd Lyn Lewis

Jan. 3rd John Jacobs

Jan. 3rd Sally Struthers

Is this a function?

Does each input have only one output?

Similarly, does each date have only one person celebrating that day as their birthday?

No. Since there is an input that has more than one output, then this is not a function.

Vertical Line Test

A relation is a function if its graph passes the vertical line test. All vertical lines must intersect the graph only once.

Are these functions? Do they pass the vertical line test?

Yes! No! No! Yes!

Evaluate the Following Function in Function Notation

f(x) = 2x – 3 when x = -2 f(-2) = 2(-2) – 3 = -4 – 3

= -7

You try! Evaluate the Following Function in Function Notation…

f(x) = -7x – 3 when x = 4 f(4) = -7(4) – 3 = -28 – 3

= -31

Graphing a Function

Graph f(x) = -1/2x + 4 Replace f(x) with y Graph y = -1/2x + 4

(0, 4)

Is the relation a function? If yes, state the domain and range.

x y

1

2

3

4

-2

-3

-3

-5

Yes, the relation is a function!

The domain is 1, 2, 3, and 4

The range is -2, -3, -3, -5

Real-world functions

Coke machine Toaster Lets come up with a few as a class.

Function

Function- exactly one output for one input.

Function Rule

Function Rule- equation that describes a functional relationship. X= total cokes 1= price per coke N= number of cokes needed

Writing a function rule

# of loads

1 2 3 4

Cost $2.75 $5.50 $8.25 $11.00

# of loads

1 2 3 4

Cost $2.75 $5.50 $8.25 $11.00

Find the difference between each box

Put into fraction form and divide

Plug into equation

Check it...

# of loads

1 2 3 4

Cost $2.75 $5.50 $8.25 $11.00

What if it doesn’t work?

TimeCost of Bike Rental

1 $10

2 $16

3 $22

4 $28

What if it doesn’t work???

TimeCost of Bike Rental

1 $10

2 $16

3 $22

4 $28

What do I need to add?

The Rule is. Check it with another number.

Time Cost

1 $10

2 $16

3 $22

4 $28