Difference between Equations and Functions

SALMAN KHAN: I'm here with
Jesse Roe of Summit Prep.

What classes do you teach?

JESSE ROE: I teach algebra,
geometry, and algebra II.

SALMAN KHAN: And now
you're with us, luckily,

for the summer, doing
a whole bunch of stuff

as a teaching fellow.

JESSE ROE: Yeah, as
a teaching fellow

I've been helping with
organizing and developing

new content, mostly on the
exercise side of the site.

SALMAN KHAN: And the reason
why we're doing this right now

is you had some very
interesting ideas or questions.

JESSE ROE: Yeah, so
as an algebra teacher,

when I introduce that concept
of algebra to students,

I get a lot of questions.

One of those
questions is, what's

the difference between an
equation and a function?

SALMAN KHAN: The difference
between an equation verses

a function, that's an
interesting question.

Let's pause it and
let the viewers

try to think about
it a little bit.

And then maybe we'll
give a stab at it.

JESSE ROE: Sounds great.

So Sal, how would you
answer this question?

What's the difference between
an equation and a function?

SALMAN KHAN: Let me think
about it a little bit.

So let me think.

I think there's
probably equations

that are not functions
and functions that

are not equations.

And then there are probably
things that are both.

So let me think of it that way.

So I'm going to draw-- if
this is the world of equations

right over here, so
this is equations.

And then over here is
the world of functions.

That's the world of functions.

I do think there
is some overlap.

We'll think it through
where the overlap is,

the world of functions.

So an equation that is not a
function that's sitting out

here, a simple one would
be something like x plus 3

is equal to 10.

I'm not explicitly talking
about inputs and outputs

or relationship
between variables.

I'm just stating an equivalence.

The expression x plus
3 is equal to 10.

So this, I think, traditionally
would just be an equation,

would not be a function.

Functions essentially
talk about relationships

between variables.

You get one or more
input variables,

and we'll give you only
one output variable.

I'll put value.

And you can define a function.

And I'll do that in a second.

You could define a
function as an equation,

but you can define a function
a whole bunch of ways.

You can visually
define a function,

maybe as a graph-- so
something like this.

And maybe I actually
mark off the values.

So that's 1, 2, 3.

Those are the
potential x values.

And then on the
vertical axis, I show

what the value of my
function is going to be,

literally my function of x.

And maybe that is 1, 2, 3.

And maybe this
function is defined

for all non-negative values.

So this is 0 of x.

And so let me just draw--
so this right over here,

at least for what I've drawn
so far, defines that function.

I didn't even have
to use an equal sign.

If x is 2, at least the way
I drew it, y is equal to 3.

You give me that input.

I gave you the value
of only one output.

So that would be a legitimate
function definition.

Another function
definition would

be very similar to what you
do in a computer program,

something like, let's say, that
you input the day of the week.

And if day is equal to Monday,
maybe you output cereal.

So that's what we're
going to eat that day.

And otherwise, you
output meatloaf.

So this would also
be a function.

We only have one output.

For any one day of
the week, we can only

tell you cereal or meatloaf.

There's no days where you
are eating both cereal

and meatloaf, which
sounds repulsive.

And then if I were to
think about something

that could be an
equation or a function,

I guess the way I think about
it is an equation is something

that could be used
to define a function.

So for example, we could say
that y is equal to 4x minus 10.

This is a potential
definition for defining y

as a function of x.

You give me any value of x.

Then I can find the
corresponding value of y.

So this is at least how
I would think about it.