Welcome to three examples of expressing a function as a composite function. Your directions may
also say "decompose a given function." Here, we’re given h of x equals the quantity two x
plus three raised to the third power. We want to write h of x as a composite function in the form
f of g of x. Remember, we can also think of this as f after g of x.
When decomposing a function, I think it's often helpful to use the alternative form of
a composite function given by this equation here. So we want h of x equal to f of g of x, which we
can also write in this form here: f of g of x. The reason this form is helpful is that, in this form,
it's easy to see that g of x is the inner function and function f would be the outer function.
So when decomposing a function, we want to be able to look at the given function
and determine which part is the inner function and which part is the outer function.
Looking at h of x, our inner function is going to be the quantity two x plus three,
and our outer function, function f, is going to be the cubing function.
So to decompose a function in the form of f of g of x, we want to define function g and function f
so that this composite function is equal to the original function. So our ultimate goal is to
find what we're going to use for function f of x and function g of x. But we already said g of x is
the inner function, so in this case, it would be the quantity two x plus three, and our function f,
the outer function, is going to be the cubing function. So f of x is equal to x cubed.
Now, function decomposition is not unique, but to me, this seems like the most obvious
decomposition. So these would be our two functions, so that f of g of x is equal to h of x.
And let's go ahead and check this. h of x
should be equal to f of g of x. Let's go and write it in this form here. Well, g of x is
equal to the quantity two x plus three, so we can write this as f of the quantity two x plus three.
So this quantity becomes the input into function f, which cubes its
input. So this does give us h of x, which is the quantity two x plus three raised to the third.
So we've decomposed this function correctly.
Let's take a look at a second example.
Same question, different function. So we want h of x equal to f of g of x,
which we'll go and write using the alternative form or this form here.
So again, g of x will be our inner function, and f will be our outer function. Looking at h of x,
let's let our inner function g of x equal the linear function five x minus one. And if we do
this, then our function f, the outer function, will be equal to the square root function.
So again, to decompose function h into f of g of x, our goal is to define function f of
x and define function g of x. As we already said, g of x is the quantity five x minus one,
our linear function, and f of x will be the square root function, or the square root of x.
This is what we're being asked to find, and let's go ahead and check this,
meaning we want to make sure h of x is equal to f of g of x.
Well, g of x is equal to the quantity five x minus one, so this is equal to f of five x minus one.
This quantity becomes the input into function f, so we replace
x with five x minus one, giving us the square root of five x minus one,
which is h of x. So that checks as well, and again, these functions are not unique,
but to me, this does seem like the most obvious composition for function h.
Let's try one more. We have h of x equals one divided by the quantity x minus five,
and the question is the same: we want to write h of x as the composite function f of g of x,
which we can also write using this notation here.
So we need to identify the inner function and outer function. This one may not be quite as
obvious. g of x is our inner function. Let's let g of x equal the denominator of a rational function;
therefore, the outer function f is just going to be the rational function one divided by its input.
So again, we have f of x, and we have g of x,
and we're saying g of x will equal the quantity x minus five,
and therefore the outer function will be this fraction function, which would just be one over
x. So these are the two functions the question is asking us to determine.
Let's go and check this to make sure our composite function is equal to h of x.
Now, we'll replace g of x with x minus five, so we'll have f of x minus five.
This becomes the input into function f, which is just one divided by its input or
one divided by, in this case, x minus five, which is equal to h of x.
And that's going to do it for this lesson. I hope this was helpful.