This video is provided as supplementary
material for courses taught at Howard Community
College and in this video I want to talk about finding the
domain of a composition of functions. Here’s the first example.
I’ve got these two functions: f(x)=x^2 and g(x)=square root of x minus 4,
and let’s say I’m told to find f compose g of x and state the domain of the composition. Now
when we have the domain of a function what we’re basically asking is what
values of x can I have which will allow the function to
produce some result. In other words, what values of x can I
place into the function? So the question
here would be, when I find out where the g compose g is,
could I have f compose g of 7,
or g compose g of 0, or f compose g of -10. Okay, so let’s first convert this f compose g into a simpler form. We’re going to make it into f(g(x)), and that’s going to let us to
replace the g(x) with what g(x) equals, which is the square
root of x minus 4. So now I’ve got f (square root of x – 4). So let’s see… Our f function says that f(x)=x^2, so f (square root x – 4) will equal the square root of x – 4 squared. Now whenever we square a square-root, we get whatever
was underneath the radical sign. In other words this just equals x – 4. So it looks like f compose g of x is going to equal x – 4. Now if I want to know what the domain of this new composed function is, I would just ask myself
“are there any numbers which x could not be?”
And it seems pretty simple that x could be just about any number I want.
It could be a positive number or a negative number. It could be 0.
But it’s not quite that simple. Even though the composition,
this final result, seems to show that x
could be any real number, from negative infinity to positive infinity,
we have to think of its parts. Especially, we have to think
about what I’m calling the inside function of this composition,
that g(x). If we look at g(x), g(x) equals the square root
of x minus 4, and I can’t use just any number for x. After all, if x was,
let’s say, 0, then the square root of 0 – 4
would be the square root of -4, and I can’t
take the square root of a negative number. So what I want to do is take this g(x) function and find out what x could be,
find out what its domain would be.
So down here, let’s take this x – 4 and
realize that it has to be either greater than or equal to 0. I can
take the square root of 0. I can take the square root of a positive number, but not a negative
number. So to solve this inequality, I’ll just add 4 to both sides.
I’m going to have x is greater than or
equal to 4. So the domain for this composition f(g(x)) equals x – 4 is going to be
restricted. It’s going to be x has to be greater than or equal to 4. So when we’re looking for the domain,
we’re doing two things: we’re finding the composition and
see if there are any restrictions on that composition.
In this case there weren’t. We’re also looking at the
individual functions, specifically the inside function, the g in this case, and seeing what the
restrictions on its domain are. We have restrictions there,
and so the restriction for the whole composition is going to be,
in this case, x has to be greater than or equal to 4. Okay, let’s take a quick look
at another one. So in this case I’ve got f(x)=square root of x and g(x)=3x + 2. And once again
let’s do f compose g of x and find out what
the composition is and what the restrictions are going to
be. So I’ll turn this into f(g(x)). Since I know what g(x) is, 3x + 2, I’ll rewrite this as just f(3x + 2). And then the f function says
take the square root of that. So that’s going to be the square root of 3x + 2. So f compose g of x
is equal to the square root of 3x + 2. Well I’ve got a square root here so there
are going to be restrictions on this. 3x + 2 is going to have to be
greater or equal to 0.
So let’s work that out. 3x + 2 is greater than or
equal to 0. I’ll subtract two from both sides.
So I have 3x is greater than or equal to -2. Now I’ll just divide both sides
of that inequality by 3 and I get x has to be
greater than or equal to -2/3. So x is greater than
or equal to -2/3.
Before I stop, let’s go back to the g(x) function and
make sure there are no restrictions there. The g(x) function was just 3x + 2, and since x could be any number here —
it could be a positive number, or 0 or a negative number,
the g(x) part of it is not going to add any more
restrictions. So now I can state that
f compose g of x equals the square root of
3x + 2 and x has to be greater than or equal to -2/3. So when you’re doing the domain
of a the composition of functions, make sure you wish look for two things.
One is what’s the domain of the final composition
and, as we saw in this other problem,
what’s the domain of the inside function, the function that we deal with first.
In this case it was the g(x) function. So that’s about it. Take care.
I’ll see you next time.