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CCSS Math: 8.F.A.1, HSF.IF.A.1

Determine whether the
points on this graph represent a function. Now, just as a
refresher, a function is really just an association
between members of a set that we call the domain
and members of the set that we call a range. So if I take any member of
the domain, let's call that x, and I give it to the
function, the function should tell me what member of
my range is associated with it. So it should point
to some other value. This is a function. It would not be a
function if it says, well, it could point to y. Or it could point to z. Or maybe it could point
to e or whatever else. This would not be a function. So this right over
here not a function, because it's not clear if
you input x what member of the range you're
going to get. In order for it
to be a function, it has to be very clear. For any input into
the function, you have to be very clear
that you're only going to get one output. Now, with that out of
the way, let's think about this function that
is defined graphically. So the domains,
the valid inputs, are the x values where
this function is defined. So for example, it tells us
if x is equal to negative 1-- if we assume that this
over here is the x-axis and this is the
y-axis-- it tells us, when x is equal to negative
1, we should output. Or y is going to be equal to 3. So one way to write
that mapping is you could say, if you
take negative 1 and you input it into
our function-- I'll put a little f box right over
there-- you will get the number 3. This is our x. And this is our y. So that seems reasonable. Negative 1 very clear
that you get to 3. Let's see what happens
when we go over here. If you put 2 into the function,
when x is 2, y is negative 2. Once again, when x
is 2 the function associates 2 for x, which
is a member of the domain. It's defined for 2. It's not defined for 1. We don't know what our
function is equal to at 1. So it's not defined there. So 1 isn't part of the domain. 2 is. It tells us when
x is 2, then y is going to be equal to negative 2. So it maps it or associates
it with negative 2. That doesn't seem too
troublesome just yet. Now, let's look over here. Our function is also
defined at x is equal to 3. Our function associates or maps
3 to the value y is equal to 2. That seems pretty
straightforward. And then we get to x
is equal to 4, where it seems like this thing
that could be a function is somewhat defined. It does try to
associate 4 with things. But what's interesting here
is it tries to associate 4 with two different things. All of a sudden in this
thing that we think might have been a
function, but it looks like it might
not be, we don't know. Do we associate 4 with 5? Or do we associate
it with negative 1? So this thing right over
here is actually a relation. You can have one member of
the domain being related to multiple members
of the range. But if you do have
that, then you're not dealing with a function. So once again, because of
this, this is not a function. It's not clear that when
you input 4 into it, should you output 5? Or should you output negative 1? And sometimes there's something
called the vertical line test that tells you whether
something is a function. When it's graphically defined
like this, you literally say, OK, when x is 4, if
I draw a vertical line, do I intersect the function
at two places or more? It could be two or more places. And if you do, that means that
there's two or more values that are related to that
value in the domain. There's two or more
outputs for the input 4. And if there are two or more
outputs for that one input, then you're not dealing
with a function. You're just dealing
with a relation. A function is a special
case of a relation. Or you could view it as
a well-behaved relation.