Evaluating with function notation | Functions and their graphs | Algebra II | Khan Academy

In this video, I want
to do a few examples

dealing with functions.

Functions tend to be something
that a lot of students find

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

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

straightforward idea.

And you sometimes wonder,
well what was all

of the hubbub about?

All a function is, is an

association between two variables.

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

means is, you give me an x.

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

You pop an x into
this function.

This function is just
a set of rules.

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

associate some value y.

You can imagine it is
some type of a box.

That is a function.

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

other number y.

This might seem a
little abstract.

What are these x's and y's?

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

Let's say I have a function
definition

that looks like this.

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

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

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

And I'm going to produce
3 otherwise.

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

So let's draw the
box around it.

This is our box.

This is just an arbitrary
function definition, but

hopefully it'll help you
understand what's actually

going on with a function.

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

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

What is f of 7 going
to be equal to?

So I take 7 into the box.

You could view it as some
type of a computer.

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

It says, OK. x is 7.

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

I go to the otherwise
situation.

So I'm going to pop out a 3.

So f of 7 is equal to 3.

So we write f of 7
is equal to 3.

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

this association, this
mapping, whatever you

want to call it.

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

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

What is f of 2?

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

give it an x equal 2.

Then the little computer inside
the function is going

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

No, I'm still in the otherwise
situation.

x isn't 0 or 1.

So once again f of
x is equal to 3.

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

Now what happens if x
is now equal to 1?

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

So f of 1.

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

Oh look, x is equal to 1.

I can use my rule right here.

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

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

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

That's all a function is.

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

problems. They tell us for
each of the following

functions, evaluate these
different functions-- these

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

different points.

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

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

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

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

do with the x?

What do I produce?

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

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

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

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

It's negative 2 times
negative 3 plus 3.

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

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

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

6 plus 3, which is equal to 9.

So f of negative 3
is equal to 9.

What about f of 7?

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

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

times 7 plus 3.

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

negative 11.

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

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

That's what this just
told us right there.

This is the rule.

This is completely analogous
to what I did up here.

This is the rule of
our function.

Let's do the next two.

I won't do part b.

You can do part b for fun.

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

Now we are at f of 0.

Here I'll just do
it in one color.

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

Wherever we see an
x, we put a 0.

So negative 2 times 0 plus 3.

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

So f of 0 is 3.

Then one last one. f of z.

They want to keep it
abstract for us.

Here I'll color code it.

So f of z.

Let me make the z in
a different color.

f of z.

Everywhere that we saw
an x, we will now

replace it with a z.

Negative 2.

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

We're going to put an
orange z there.

Negative 2 times z plus 3.

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

If you imagine our box,
the function f.

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

whatever that z is plus 3.

That's all this is saying.

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

Now let's just do part c here.

Let me clear this actually.

I'm running out of
real estate.

Let me clear all of
this business.

Let me clear all of
this business.

We can do part c.

I'm skipping part b.

You can work on that part.

Part b.

They tell us-- this is our
function definition.

Sorry, I said I was
doing part c.

This is our function
definition.

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

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

inputs into our function.

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

see an x, we put a negative 3.

2 minus negative 3 over 11.

This is equal to 2 plus 3.

This is equal to 5.

So you get 5 times 5 over 11.

That's equal to 25/11.

Let's do this one.

F of 7.

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

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

2 minus 7 over 11.

So what is this going
to be equal to?

2 minus 7 is negative 5.

5 times negative 5 is
negative 25/11.

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

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

5 times 2 is 10.

So this is equal to 10/11.

One more.

f of z.

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

replace it with a z.

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

And that's our answer.

We could distribute the 5.

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

We could even write it in
slope-intercept form.

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

These are all equivalent.

But that is what f
of z is equal to.

Now.

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

give you an output.

I will give you an f of x.

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

produce an f of x.

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

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

values for an x.

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

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

x is equal to 0.

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

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

What it's going to be equal?

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

we don't know.

We don't know.

We don't know.

This is not a function
even though it might

have looked like one.

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

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

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

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

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

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

is my y value here.

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

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

is for a certain x value.

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

So this is a valid function.

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

the graph once.

So this is a valid function.

Now what about this
one right here?

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

that point right there.

For that x, this relation
seems to have two

possible f of x's.

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

Right?

We're intersecting
the graph twice.

So this is not a function.

We're doing exactly what
I described here.

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

that could be equal to f of x.

So this is not a function.

Over here, same thing.

You draw a vertical
line right there.

You're intersecting
the graph twice.

This is not a function.

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

Let's go to this function.

It's kind of a weird
looking function.

Looks like a reversed
check mark.

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

intersecting it once.

So this is a valid function.

For every x, you only have
one y associated.

Or only one f of x associated
with it.

Anyway, hopefully you
found that useful.