Relations and Determining Whether a Relation is a Function - Concept
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
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
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.
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.