Evaluate a Function at Given Values

>> In this video,
we're going to look at

evaluating a function
for a given value.

I've got this function up here.

F(x) = -4x^2-7x+9.

The first thing that you really
want to make sure you're

comfortable with is the
notation for functions.

This, I read this

f(x) the first time you
see function notation,

it looks like f*x,

where f is some variable and
x is some other variable.

But that is totally not
what's happening here at all.

What we have is f(x).

We have a function f and

it's going to be evaluated
at a variable x.

In other words, we're going
to take x and plug it

in to this equation over here.

A lot of times when I'm
first teaching functions,

I'll remind the students that

f(x) is really the
same as y. I could

rewrite this whole thing as

y = -4x^2-7x+9 and it's
basically the same equation.

This is function notation
up here in black

and this is just written with

two different
variables here in red,

but it's the same exact thing.

Let's say we want to evaluate
this function at 1/3.

In function notation, I
would write it like this.

I would say find f(1/3).

What that means is plug
1/3 in for x up here.

If I was doing it with the
equation in two variables,

I would say find y when x = 1/3.

If I gave you that
equation and told

you to find y when x = 1/3,

you just take 1/3 and plug
it in for all that stuff.

If I give you this
function notation

until you to find f(1/3),

then you just plug 1/3 in
where all the x's are.

I'm going to rewrite
this equation.

Wherever there's an x,

I'm going to put a 1/3.

There's two spots
I need to put 1/3.

Now from this point,

it's just a matter of arithmetic
in order of operations.

It's nothing fancy.

It's just working
the problem out.

As a reminder, let's write
our order of operations,

parentheses or grouping symbols,

exponents, multiplication,
division, those are equals.

We do them left to right
and addition, subtraction.

Those guys are equals. We
do them left to right.

Addition does not always
come before subtraction

and multiplication does not
always come before division,

it depends which ones first.

In this problem, it looks

like we have some
parentheses here.

A lot of people say,
well I have parentheses,

but parentheses really means to

simplify what's inside
the parentheses when

we're talking about the order
of operations and this is

already simplified
inside the parentheses.

The parentheses in
this case are really

just standing for
multiplication.

The next thing we're going
to look at is exponent's.

We do have an exponent
here, (1/3)^2.

We're going to
bring down the -4.

We're going to do (1/3)^2.

Now I'm going to keep
these parentheses

because in this case it
means multiplication.

I still need to multiply
my answer by -4,

1^2 is 1 and 3^2 is 9.

That's as easy as it is
for squaring fractions.

A lot of people think
that's tough. It's not.

You just square the top,

square the bottom, you're done.

I'll bring the rest of this down

and we'll go to the next step,

which would be
multiplication or division.

Don't have any division.

Technically, a fraction is

a division problem,
but if you divide it,

you're going to
get a decimal and

we're not putting
these in decimals,

we're going to keep
it in a fraction.

We're not going to
do that division,

we're going to do
the multiplication.

We're going to take -4*1/9.

Some of you are
really good at this

and already have the
answer in your head.

Those of you out there that are

still struggling with
fractions a little bit.

Remember when you
multiply fractions,

you just multiply the
numerators and denominators.

If you have a whole number
like four, -4 is an integer.

Positive four is a whole
number technically.

But if you have a number that's

not a fraction like an integer,

like -4, that's
the same as -4/1.

Sometimes it helps to
write that in there.

Seven is the same as 7/1.

Once you get used to that,
you don't have to write

that over one stuff
because you can

see it in your head that
you're going to multiply

the -4*1 is -4,

1*9 is 9.

With the -4 or you can think of

it out in front like -4/9.

That's probably the best
way to think of it.

At this point, is
-4/9. Over here.

Then we're going to
bring down our -7/3+9.

Well, at this point we need
to get a common denominator.

Let's get a common
denominator of nine.

You know what I'm going
to do? I'm going to

do these two fractions first.

A lot of times when you have

just adding a whole
number at the end,

you can do it in your head.

Let's just do the fractions
and see what we've got.

I'm going to multiply
this by 3/3 to

change it to 1/9 and bring

the -4/9 down, -7*3 is (21/9)+9.

When you're adding or
subtracting fractions,

you add or subtract the
numerators, -4-21 is (-25/9)+9.

Now some of you guys might
want to change this nine to

81/9 and get a
common denominator.

That's fine. I'm going to
go ahead and finish it

this way. Negative 25/9.

Now I'm going to divide, I
have an improper fraction,

so nine goes into 25,

2 times, -2 and how
many are leftover?

Let's see 18, so
that'd be seven.

I've got this now,

-2 (7/9)+9, which is really
the same as 9-2 7/9.

That's the same thing.

Nine takeaway 2 would be 7,

take away 7/9 more
would be 6 2/9.

If I lost you there,

let me show you this other
way that you can do it.

Let's go this way
with the blue arrow.

I could keep the 25/9 and then
change the nine into 1/9,

which would be 81/9, 81/9 is 9.

Then do -25+81 on the top,

which is the same as 81
takeaway 25, which is 56/9.

Of course, 56/9 is
the same as 6 2/9.

You could do it either
way with the fractions.

Whichever way makes more sense

to you is the way that
you should do it.

But back to the main concept of

evaluating a function
for a given value.

It's just a matter of
plugging in whatever number

is in there for x that's
going to go where the xs are.

You going to put that
number in for x,

whatever is in this parentheses

that goes in where the xs are.

Then just make sure you
follow your order of

operations and watch
out for those things

like positives and negatives and

fractions and denominators
and stuff like that,

then it becomes a
problem from your past.

Hope that helps and makes

functions a little
easier to understand.