Intro To Functions

Module 4, Lesson 1

Online Algebra

VHS@PWCS

Graphing review

If you cannot remember how to graph points on a coordinate plane from Pre-

Algebra, then you can do one of 2 things or both!

Review page 58 in your textbook. Look at the video title graphing review

This is an important skill that we will use a lot from hear on out.

Relations

In algebra a relation is a set of ordered pairs. They do not have to have a rule, but can

be any set of ordered pairs. Because we are talking about sets, we

need to use braces { }. Braces mean all of these things are in a set.

An example of a relation is below:

{(3, 2), (0, 0), (4, 9), (-3, -7)}

Different Ways To Show Relations.

One of the ways to show the relation {(4, 5), (4, 2), (-1, -1), (2, -1)} is a mapping.

X – coordinates Y – Coordinates

-1

2

4

-1

2

5

Each x- value MAPS to a y-value.

The arrows show us the ordered pairs.

Notice that x-coordinate of 4 has two y – values (2 and 5), but 4 is used once, but the two arrows indicate the different ordered pairs.

Different Ways To Show Relations

Two other ways to show the relation {(4, 5), (4, 2), (-1, -1), (2, -1)} are a table and a graph.

X Y

-1 -1

2 -1

4 5

4 2

Graph:

(-1, -1) (2, -1)

(4, 2)

(4, 5)

Table:

Domain and Range

A relation is made up of domain and a range.

The Domain is the set of all x-coordinates. Other names for the domain are independent

variables and input. The Range is the set of all y-coordinates.

Other names for the range are dependent variables and output.

Domain and Range

X Y

-1 -1

2 -1

4 5

4 2

What is the domain and range of the following relation?

Domain: {-1, 2, 4}•Notice that even though there are two fours, we only state it once. We also used braces because the domain is a set.

Range: {-1, 2, 5}•Notice that even though there are two negative ones, we only state it once. We also used braces because the range is a set.

Relation Review

A relation is a set of ordered pairs. The can be displayed 4 different ways

Set of ordered pairs Mapping Table Graph

A relation is made up of a domain and range. The domain is all the x-coordinates

Other names for the domain are: independent variables and input The range is all the y-coordinates.

Other names for the range are: dependent variables and output.

Functions

A function is a special relation. A function assigns exactly one value of

the dependent variable to each value of the independent variable. In other words each x – coordinate maps

to only 1 y-coordinate.

How To Tell if a Relation is a Function.

In a table or set of ordered pairs, if an x-value repeats then it is not a function.

X Y

-1 -1

2 -1

4 5

4 2

Notice the 4 repeats, so it is not a function!

X Y

-1 -1

2 -1

6 -5

10 2

In this relation, no x-values repeat, so this is a function.

It is okay for y-values to repeat.

In a mapping each x, should have only 1 arrow coming from it.

How To Tell if a Relation is a Function.

X

2

4

6

Y

-2

-1

This is a function X

2

4

Y

-2

-1 This is not a function

Different Ways To Show Relations

In a graph, we use a vertical line test. If the graph passes through the vertical line more than once it is not a function.

The vertical line goes through the graph at 2

places it is not a function.

Here no matter where we draw a vertical line, that line will only go through our graph once. So this is a function.

Which of the following are functions?

{(0, 2), (1, 2) (3, 2)}

x 1 1 3

y -2 4 3

x

3

4

Y

-1

3

Only the set of ordered pairs is a function, in both the table and the mapping there is an x

that repeats and the graph fails the vertical line test.

Function Review

Remember that a function is a special relation where each member of the domain maps to exactly one member of the range.

Other names for the domain are: x-coordinates, input, independent variable

Other names for the range are: y-coordinates, output, dependent variable.