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Relations and Determining Whether a Relation is a Function - Concept
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Understanding relations (defined as a set of inputs and corresponding outputs) is an important step to learning what makes a function. A function is a specific relation, and **determining whether a relation is a function** is a skill necessary for knowing what we can graph. Determining whether a relation is a function involves making sure that for every input there is only one output.

One of the things you guys learn in

your math classes is that there's

ton of vocabulary to keep straight

in your head.

So we're going to look at a few vocabulary

words before we start looking at numbers.

First is the domain. The domain is the

set of all X. Sometimes called

input values.

The range is the set of

all Y or output values.

That will make more sense as we start

looking at like actual numbers.

In order for a relation to be called a function,

each X value must have exactly one Y value.

Function is a really important word in math

class, and we're going to practice

that more and more.

So let's start looking at some actual numbers

where this will make more sense.

Before we do that, keep in mind each

X has to have exactly one Y value.

You can't have two Y values and

you can't have no Y values.

So that's something to keep in mind.

It has to have exactly one Y value.

So let's look at a couple of examples.

A lot of times you guys see math information

organized in a table.

Here I have my X numbers,

8, 9, 10 and 13.

That's kind of weird.

Don't be freaked out.

Sometimes math numbers are consecutive,

like 8, 9, 10.

Sometimes there's a wildcard

thrown in there like 13.

Don't worry, it's okay.

Still going to work.

That's my domain, 8, 9, 10, 13.

My range is my Y numbers.

Negative 1, negative 3, 5. Let me show you

another way you might see this written.

And that's using ordered pairs.

And you guys have seen ordered pairs before

when you first learn how to graph.

It's like a point, right?

So I could write 8 comma negative 1 and

that would represent X equals 8, Y is

negative 1. I'm going to go through

and write all of these in the

point notation.

And then one thing that's kind of weird

is we use these little curly brackets.

This is called set notation.

And you'll get into this

more in your future.

And then the last way you might see this

is through what's called a map or a mapping.

And I'm going to draw these little bubbles.

1 is going to represent

my Xs or my domain.

The other is going to represent

my Ys or my range.

So I have the X numbers

8, 9, 10 and 13.

I have the Y numbers negative 1, negative

3 and 5. Notice that I didn't

write negative 1 twice.

Even though negative 1 shows up twice in

the table and shows up twice in the

ordered pairs, it only shows

up once in the mapping.

And the thing I like about this is it's

kind of easy to see what number goes

with which when you draw arrows.

Like 8 arrow to negative 1. 9 goes

to negative 3. 10's also going

to go to negative 1 so I'm going to

have that double arrow coming into

negative 1 and then 13 is

matched with 5. Okay.

So those are all different ways to

represent the same relationship.

Now I want to think about is whether

or not this is a function.

Keep in mind in order to be a function each

X value must have exactly one Y value.

So go through and look.

Does each X have a Y? Yes.

You want to make sure that no

X number shows up twice.

In our case this is a function, because

each X number has exactly one Y number,

but watch this.

If I were to stick in 8 again with something

that's not negative 1, like 8 and,

I don't know, 15 or something,

this would not be a function.

Because now 8 has two different Y values.

Let me get rid of that because nonfunctions

kind of freak me out.

Okay.

Important thing to keep me in mind is that

each X number has to have exactly

one Y number.

And that will be called set of X is the

domain and we call the set of Yes the range.

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