We're asked to find f of g of h of x
and determine the domain.
Looking at the composition of three functions,
notice how the innermost function is h of x
where h of x is equal to the square root
of the quantity x plus 3,
which means f of g of h of x
is equal to f of g
of the square root of the quantity x plus 3.
Notice now here we substituted the square root
of the quantity x plus 3 for h of x.
And now the square root of the quantity x plus 3
becomes the input into g of x,
where g of x is equal to 1 divided
by the quantity x squared minus 2,
which means f of g
of the square root
of the quantity x plus 3 is equal to f of
1 divided by the square
of the square root of the quantity x plus 3 minus 2.
Let's go ahead and simplify this fraction.
If we square the square root of the quantity x plus 3,
we just get x plus 3,
which gives us f of 1
divided by the quantity x plus 3.
And then we still have minus 2,
which gives us f of 1 divided by the quantity x plus 1.
So notice how 1 divided by the quantity x plus 1
is g of h of x.
And now finally,
to find f of 1 divided by the quantity x plus 1,
we replace the x in f of x with 1 divided
by the quantity x plus 1.
This gives us the square of 1 divided
by the quantity x plus 1, and then plus 5.
So this may be one way the composition
of functions is expressed.
Let's also square the fraction.
To square a fraction,
we square the numerator and we square the denominator.
So we can also express the composition of functions as f
of g of h of x
equals the square of 1 divided by the square of x plus 1
plus 5.
Well, the square of 1 is 1.
And if we square x plus 1, there are no shortcuts.
We have two factors of x plus 1,
giving us 4 products: 1, 2, 3, 4,
which gives us x squared plus x plus x,
which gives us plus 2x and then plus 1.
Giving us 1 divided by the quantity x squared
plus 2x plus 1.
And of course, we still have plus 5.
So the composition of functions can also be expressed
in this form.
And that's what going to determine the domain
of the composition of 3 functions,
which can be a little bit tricky.
So we will follow the steps below.
Step 1: We will find the domain restriction
of the function rule h of x.
Step 2: We'll find the domain restriction
of the function rule g of h of x.
And then finally, step 3:
We'll find the domain restriction
of the function rule f of g of h of x.
From here the domain of the composition of functions
must contain all the restrictions from 1 through 3.
So let's begin by determining the domain restriction
for h of x where h of x is equal to the square root
of the quantity x plus 3.
The radicand, or the number under the square root,
must be now negative,
which means x plus 3 must be greater than
or equal to zero.
So if x plus 3 must be greater than or equal to zero,
if we solve for x, we subtract 3 on both sides,
which gives us x is greater than or equal to negative 3.
Using interval notation,
this would be the interval where the interval is closed
on negative 3 to positive infinity.
And now we need to find the domain restriction
for the function rule g of h of x.
Remember g of h of x would be the composition
of the 2 innermost functions,
which we found as 1 divided by the quantity x plus 1.
And since division by zero is undefined,
x plus 1 can't be zero.
If we solve x plus 1 can't equal zero, for x,
we get x can't equal negative 1.
Using interval notation,
this would be the open interval
from negative infinity to negative 1 union,
the open interval from negative 1 to infinity.
And now let's consider the function rule
for f of g of h of x
and let's go and use this form
of the composition of functions here.
Once again, when your division by zero is undefined
and therefore x plus 1 can't equal zero,
meaning x can't equal negative 1.
So notice now,
this restriction is the same as the restriction
for g of h of x.
Using interval notation,
this is the open interval from negative infinity
to negative 1 union,
the open interval from negative 1 to positive infinity.
So now to determine the domain
of the composition of functions,
we need to consider all three of these restrictions,
but notice how the restrictions
for the last two function rules are the same.
So let's focus on the restrictions
for the first two function rules.
Which means we need to determine the intersection
of the intervals expressed in interval notation.
Working our way from left to right along the number line,
let's begin by considering the intersection
of these two intervals here.
Notice, the interval that satisfies both conditions
would be the interval that's closed on negative 3
and open on negative 1.
Let's go ahead and record this.
From here, the open interval from negative 1
to positive infinity is in the interval
that's closed on negative 3 to positive infinity.
And therefore we also include the open interval
from negative 1 to infinity.
This interval satisfies all the restrictions
and therefore is a domain
of the composition of functions f of g of h of x.
Let's also express this using inequalities.
We could also say
that x is greater than or equal to negative 3
and less than negative 1,
or x is greater than negative 1.
I hope you found this helpful.