Composite functions

Composite functions are, as the name says a bit, when we put functions together. By "put together" we mean that we put one function into another. The hurdle is the order: putting $g$ into $f$ is not the same as putting $f$ into $g$.

When do I use this?

Whenever a question gives you two functions and asks for $fg(x)$, $gf(x)$, $fg(3)$ or something similar. You need function notation to be second nature first, because a composite function is nothing more than function notation used twice.

The procedure

Let's start with an example. I have two functions:

$$f(x) = x^2 \qquad g(x) = 2x$$

If I put $g(x)$ into $f(x)$, I just have to take the whole function $g(x)$ and put it in x's place in the other function, $f(x)$, like this:

$$(2x)^2$$

What we've done is just to put $g(x)$ in x's place in $f(x)$. If we did it the other way round, that is, put $f(x)$ into $g(x)$, it would look like this:

$$2x^2$$

These are composite functions. We just stuff one function into the other function.

How we write it

When we write composite functions, we do it like this:

$$f(g(x))$$

This says that we've put $g(x)$ into $f(x)$. The other way round would be

$$g(f(x))$$

which means that we've put $f(x)$ into $g(x)$. We say it as "f of g of x", or the other way round: "g of f of x".

The GCSE papers use a shorthand for exactly the same thing:

$$fg(x) = f(g(x)) \qquad gf(x) = g(f(x))$$

So when you see $fg(x)$, read it as $f(g(x))$: $g$ is the inner function, $f$ is the outer one, and you do the inner one first. It's just the brackets left out to save ink.

Worked example

Let's try taking one more example. We have the functions

$$f(x) = \sin(x) \qquad g(x) = x^2$$

where $\sin$ is the same sine you know from trigonometry.

If we had to find $f(g(x))$, that is, when we put $g(x)$ into $f(x)$, we get

$$\sin(x^2)$$

because we've just put $g(x)$ in x's place in $f(x)$.

If we do it the other way, that is, find $g(f(x))$, we have to put $f(x)$ in x's place in $g(x)$, like this:

$$\sin(x)^2$$

Two different functions from the same two ingredients. That's the whole point of keeping the order straight.

Common mistakes

• "f(g(x)) is the same as g(f(x))." It isn't. In our first example $f(g(x)) = (2x)^2$ while $g(f(x)) = 2x^2$. Which function is inside decides everything.
• "I only put part of the inner function in." You have to take the whole of the inner function and put all of it in x's place. Putting $g(x) = 2x$ into $f(x) = x^2$ gives $(2x)^2$, brackets and all.
• "fg(x) means f times g." It means $f(g(x))$: $g$ first, then $f$. It's function notation, not multiplication.

Related

• What is a function?
• Function notation: what f(x) means
• Inverse functions