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In this video, I want
to do a few examples

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dealing with functions.

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Functions tend to be something
that a lot of students find

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difficult, but I think if you
really get what we're talking

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about, you'll see that it's
actually a pretty

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straightforward idea.

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And you sometimes wonder,
well what was all

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of the hubbub about?

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All a function is, is an

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association between two variables.

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So if we say that y is equal to
a function of x, all that

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means is, you give me an x.

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You can imagine this function
is kind of eating up this x.

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You pop an x into
this function.

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This function is just
a set of rules.

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It's going to say, oh,
with that x, I

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associate some value y.

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You can imagine it is
some type of a box.

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That is a function.

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When I give it some number
x, it'll give me some

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other number y.

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This might seem a
little abstract.

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What are these x's and y's?

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Maybe I have a function-- let
me make it like this.

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Let's say I have a function
definition

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that looks like this.

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For any x you give me, I'm going
to produce 1 if x is

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equal to-- I don't know-- 0.

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I'm going to produce 2
if x is equal to 1.

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And I'm going to produce
3 otherwise.

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So now we've defined what's
going on inside of the box.

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So let's draw the
box around it.

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This is our box.

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This is just an arbitrary
function definition, but

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hopefully it'll help you
understand what's actually

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going on with a function.

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So now if I make x is equal to--
if I pick x is equal to

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7, now what is f of x going
to be equal to?

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What is f of 7 going
to be equal to?

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So I take 7 into the box.

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You could view it as some
type of a computer.

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The computer looks at that x and
then looks at its rules.

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It says, OK. x is 7.

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Well x isn't 0. x isn't 1.

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I go to the otherwise
situation.

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So I'm going to pop out a 3.

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So f of 7 is equal to 3.

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So we write f of 7
is equal to 3.

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Where f is the name of this
function, this rule system, or

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this association, this
mapping, whatever you

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want to call it.

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When you give it a 7,
it'll produce a 3.

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When you give f a 7,
it'll produce a 3.

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What is f of 2?

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Well, that means instead of x
is equal to 7, I'm going to

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give it an x equal 2.

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Then the little computer inside
the function is going

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to say, OK, let's see,
when x is equal to 2.

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No, I'm still in the otherwise
situation.

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x isn't 0 or 1.

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So once again f of
x is equal to 3.

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So, this is f of 2 is
also equal to 3.

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Now what happens if x
is now equal to 1?

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Well then it's just going
to turn over this.

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So f of 1.

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It's going to look at its
rules right here.

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Oh look, x is equal to 1.

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I can use my rule right here.

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So when x is equal to
1, I spit out a 2.

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So f of 1 is going
to be equal to 2.

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I spit out f of 1, which is
equal to 2 in that situation.

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That's all a function is.

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Now, with that in mind, let's
do some of these example

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problems. They tell us for
each of the following

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functions, evaluate these
different functions-- these

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are the different boxes they've
created-- at these

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different points.

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Let's do part a first. They're
defining the box.

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f of x is equal to negative
2x plus 3.

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They want to know what happens
when f is equal to negative 3.

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Well f is equal to negative 3,
this is telling me what do I

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do with the x?

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What do I produce?

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Wherever I see an x, I replace
it with the negative 3.

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So it's going to be equal
to negative 2.

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Let me do it this way, so you
see exactly what I'm doing.

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That negative 3, I'll do
it in that bold color.

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It's negative 2 times
negative 3 plus 3.

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Notice wherever there was an
x, I put the negative 3.

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So I know what the black box
is going to produce.

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This is going to be equal to
negative 2 times negative 3 is

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6 plus 3, which is equal to 9.

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So f of negative 3
is equal to 9.

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What about f of 7?

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I'll do the same thing one more
time. f of-- I'll do 7 in

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yellow-- f of 7 is going to
be equal to negative 2

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times 7 plus 3.

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So this is equal to negative 14
plus 3, which is equal to

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negative 11.

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You put in-- let me make it very
clear-- you put in a 7

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into our function f here and it
will pop out a negative 11.

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That's what this just
told us right there.

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This is the rule.

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This is completely analogous
to what I did up here.

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This is the rule of
our function.

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Let's do the next two.

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I won't do part b.

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You can do part b for fun.

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I'll do part c after that, just
for the sake of time.

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Now we are at f of 0.

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Here I'll just do
it in one color.

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I think you're getting
the idea. f of 0.

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Wherever we see an
x, we put a 0.

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So negative 2 times 0 plus 3.

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Well, that's just
going to be a 0.

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So f of 0 is 3.

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Then one last one. f of z.

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They want to keep it
abstract for us.

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Here I'll color code it.

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So f of z.

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Let me make the z in
a different color.

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f of z.

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Everywhere that we saw
an x, we will now

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replace it with a z.

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Negative 2.

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Instead of an x, we're going
to put a z there.

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We're going to put an
orange z there.

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Negative 2 times z plus 3.

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And that's our answer. f of
z is negative 2z plus 3.

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If you imagine our box,
the function f.

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You put in a z, you are going to
get out a negative 2 times

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whatever that z is plus 3.

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That's all this is saying.

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It's a little bit more abstract,
but same exact idea.

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Now let's just do part c here.

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Let me clear this actually.

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I'm running out of
real estate.

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Let me clear all of
this business.

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Let me clear all of
this business.

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We can do part c.

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I'm skipping part b.

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You can work on that part.

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Part b.

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They tell us-- this is our
function definition.

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Sorry, I said I was
doing part c.

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This is our function
definition.

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f of x is equal to 5 times
2 minus x over 11.

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So let's apply these different
values of x, these different

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inputs into our function.

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So f of negative 3 is equal to
5 times 2 minus-- wherever we

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see an x, we put a negative 3.

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2 minus negative 3 over 11.

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This is equal to 2 plus 3.

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This is equal to 5.

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So you get 5 times 5 over 11.

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That's equal to 25/11.

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Let's do this one.

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F of 7.

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For this second function right
here, f of 7 is equal to 5

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times 2 minus-- now
for x we have a 7.

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2 minus 7 over 11.

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So what is this going
to be equal to?

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2 minus 7 is negative 5.

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5 times negative 5 is
negative 25/11.

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Then finally, well we have
two more. f of 0.

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That's equal to 5 times 2 minus
0 So this is just 2.

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5 times 2 is 10.

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So this is equal to 10/11.

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One more.

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f of z.

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Well every time we saw
an x, we're going to

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replace it with a z.

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It's equal to 5 times
2 minus z over 11.

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And that's our answer.

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We could distribute the 5.

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You could say this is the same
thing as 10 minus 5z over 11.

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We could even write it in
slope-intercept form.

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This is the same thing as
minus 5/11 z plus 10/11.

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These are all equivalent.

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But that is what f
of z is equal to.

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Now.

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A function, we said, if you give
me any x value, I will

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give you an output.

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I will give you an f of x.

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So if this is our function,
you give me an x, it will

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produce an f of x.

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It can only produce 1
f of x for any x.

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You can't have a function that
could produce two possible

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values for an x.

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So you can't have a function--
this would be an invalid

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function definition-- f
of x is equal to 3 if

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x is equal to 0.

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Or it could be equal to
4 if x is equal to 0.

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Because in this situation, we
don't know what f of 0 is.

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What it's going to be equal?

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It says if x is equal to 0, it
should be 3 or it could be--

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we don't know.

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We don't know.

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We don't know.

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This is not a function
even though it might

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have looked like one.

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So you can't have two f of
x values for one x value.

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So let's see which of these
graphs are functions.

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To figure that out, you could
say, look at any x value

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here-- pick any x value-- I have
exactly one f of x value.

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This is y is equal to
f of x right here.

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I have exactly only one--
at that x, that

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is my y value here.

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So you could have a vertical
line test, which says at any

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point if you draw a vertical
line-- notice a vertical line

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is for a certain x value.

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That shows that I only have
one y value at that point.

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So this is a valid function.

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Any time you draw a vertical
line, it will only intersect

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the graph once.

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So this is a valid function.

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Now what about this
one right here?

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I could draw a vertical
line, let's say, at

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that point right there.

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For that x, this relation
seems to have two

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possible f of x's.

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f of x could be that value or
f of x could be that value.

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Right?

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We're intersecting
the graph twice.

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So this is not a function.

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We're doing exactly what
I described here.

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For a certain x, we're
describing two possible y's

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that could be equal to f of x.

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So this is not a function.

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Over here, same thing.

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You draw a vertical
line right there.

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You're intersecting
the graph twice.

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This is not a function.

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You're defining two possible
y values for 1 x value.

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Let's go to this function.

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It's kind of a weird
looking function.

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Looks like a reversed
check mark.

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But any time you draw a vertical
line, you're only

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intersecting it once.

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So this is a valid function.

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For every x, you only have
one y associated.

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Or only one f of x associated
with it.

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Anyway, hopefully you
found that useful.