Operations on Functions - Grade 11 (General Mathematics)

Hi guys, it's me teacher Gon.

For today's video, we will do
operations on functions.

This topic is considered as
the algebra of functions,

So let's start.

Functions can be added,
subtracted, multiplied, and divided.

Such procedures are called
"operations of functions" or

"algebra of functions."

Here inside the box
are the different operations

can be used in functions

and here are the functions of
the different operations.

We have f plus g of x that is equal to

f of x plus
g of x.

We have here the second one,

f minus g of x is equal to
f of x minus g of x.

Third one, we have
f times g of x

is equal to f of x times g of x.

The fourth one: f divided by
g of x is equal to

f of x over g of x.
For addition of functions: given

the functions, f of x and g of x,

their sum is denoted by
f plus g of x.

So we will use the formula

f plus g of x is equal
to f of x plus g of x.

And now for the subtraction of functions,
given two functions,

f of x and g of x,

Their difference is denoted by
f minus g of x

is the function defined by
f minus g of x is equal to

f of x minus g of x.
So let's have an example

on how to do addition and
subtraction of functions.

So, we have here:
Given the functions

f of x that is equal to 4x (squared) plus
3x plus two, and the second function

which is g of x is equal to 7x squared
minus 5x minus one.

So, in this given example,
we are asked to find

f plus g of x, meaning we
need to add the two functions

and then we have f minus g of x, where
we're needing to get the difference

of the two functions.
So let's get started.

So, start with addition of functions.
We have to find f plus g of x

wherein your f of x is equal to
this one, so that is equal to

f of x plus g of x.
You need to substitute the value of

f of x. The value of f of x simply is
4x squared plus 3x plus two.

Include that in the parentheses. Plus
the value of your g of x.

For the expression, we have
7x squared minus 5x minus one.

And then all you need to do here is
to eliminate first the parentheses,

and then combine like terms. So we
have here the like terms, which is

4x squared and 7x squared,
we have 11x squared.

And then 3x plus negative 5x, that will
give you negative 2x.

And then for the constants, we have
two plus negative one,

that will give you plus one.

So the sum of the two functions
f of x and g of x, simplified,

is 11x squared minus 2x plus one.

Now let's move on to find
f minus g of x

wherein we need to do
subtraction of functions.

So the formula that we're going to
use is simply

f minus g of x.
And then your f of x is

4x squared plus 3x plus two
minus your g of x

is 7x squared minus 5x
minus one.

And this time we will multiply
the negative sign to the

7x squared minus 5x minus one.

So we will have the new equation,
4x squared plus 3x plus 2.

This one is negative: negative 7x squared,

this will be positive: plus 5x,

and this is also positive.
And the last thing you need to do

is to combine like terms. So your
f minus g of x

when you combine like terms, we have
4x squared and then negative 7x squared,

we have negative 3x squared, then
3x plus 5x, that will give you 8x,

two plus one is three. Therefore,
the difference between f of x and g of x

simplified is negative 3x squared plus
8x plus three. So that's it on

how to do addition.
Once we are done with addition and

subtraction of functions, let's move on to
multiplication and division of functions.

Given the functions f of x and g of x,
their product, denoted by f times g of x

is the function defined by
f times g of x is equal to

f of x times g of x.
This formula inside the box

is the formula that we're going to use
in multiplying functions.

Now, for the division of functions,
given the functions f of x and g of x

the quotient, denoted by f over g of x,
is the function defined by the formula

f over g of x is equal to f of x over g of
x again, or in g of x, or your divisor,

is not equal to 0.

Now let's go look at examples with
regards to multiplication and division of

functions. We have here:
Given the functions f of x is equal to

x squared minus one and g of x is equal
to x plus one, we are asked to find

f times g of x and f over g of x.

So let's start with the multiplication
of functions.

So we will use the formula f times g of x
is equal to f of x times g of x

wherein in this given example you need to
multiply first. Let's get the value of

f of x which is x squared minus one...
x squared minus one.

And then multiply by the value of g of x
which is x plus one. As you can see,

we have two binomials, therefore we're
going to use FOIL method.

So, let's multiply. We have x squared
times x, that will give you x cubed,

and then x squared times one, that is
plus x squared, and then

inner terms, negative one times x,
that'll give you minus x,

and the last terms, negative one times
one, that is negative one. Therefore,

the product of the function f of x
and g of x is simply...

x cubed plus x squared minus x minus one.

Now let's proceed to the division
of functions.

The formula we're going to use is f over g
of x is equal to f of x over g of x.

So let's substitute the value of f of x
which is x squared minus one

over the value of your g of x, which is
nonzero,

we have x plus one. So how you're going
to simplify f of x over g of x,

as you can see, our numerator is x squared
minus one, and then your denominator is

x plus one. We can apply factoring because
the numerator has the pattern of

difference of two squares, so we can
factor out your numerator as

x plus one times x minus one, so
that is over x plus one.

So we have your x plus one in the
numerator and x plus one in the

denominator, so we can cancel all
this out, and the remaining expression

in your numerator simplified is
x minus one. So this the quotient of

the function f of x and g of x. If you
have any questions about this topic,

you can comment down below and ask
for clarifications. So, by the way,

we will upload another video of this one
wherein the content of the whole video

is more on the examples of operations
on functions.