Composition of Functions

"Function Composition" is applying one function to the results of another:

The result of f() is sent through g()

It is written: (g º f)(x)

Which means: g(f(x))

Example: f(x) = 2x+3 and g(x) = x²

"x" is just a placeholder, and to avoid confusion let's just call it "input":

f(input) = 2(input)+3

g(input) = (input)²

So, let's start:

(g º f)(x) = g(f(x))

First we apply f, then apply g to that result:

(g º f)(x) = (2x+3)²

What if we reverse the order of f and g?

(f º g)(x) = f(g(x))

First we apply g, then apply f to that result:

(f º g)(x) = 2x²+3

We got a different result!

So be careful which function comes first.

Symbol

The symbol for composition is a small circle: (g º f)(x)

It is not a filled in dot: (g · f)(x), as that would mean multiply.

Composed With Itself

You can even compose a function with itself!

Example: f(x) = 2x+3

(f º f)(x) = f(f(x))

First we apply f, then apply f to that result:

(f º f)(x) = 2(2x+3)+3 = 4x + 9

You should be able to do this without the pretty diagram:

(f º f)(x) = f(f(x)) = f(2x+3) = 2(2x+3)+3 = 4x + 9

Domains

It has been easy so far, but now you must consider the Domains of the functions.

The domain is the set of all the values that go into a function. The function must work for all values you give it, so it is up to you to make sure you get the domain correct!

Example: the domain for √x (the square root of x)

You cannot have the square root of a negative number (unless you use imaginary numbers, but we aren't), so we must exclude negative numbers:

The Domain of √x is all non-negative Real Numbers

On the Number Line it looks like:

Using set-builder notation it is written:

{ x | x ≥ 0}

Or using interval notation it is:

[0,+∞)

It is important to get the Domain right, or you will get bad results!

Domain of Composite Function

You must get both Domains right (the composed function and the first function used).

When doing, for example, (g º f)(x) = g(f(x)):

• Make sure you get the Domain for f(x) right,
• Then also make sure that g(x) gets the correct Domain

Example: f(x) = √x and g(x) = x²

The Domain of f(x) = √x is all non-negative Real Numbers

The Domain of g(x) = x² is all the Real Numbers

The composed function is:

(g º f)(x) = g(f(x)) = (√x)² = x

Now, "x" would normally have the Domain of all Real Numbers ...

... but because it is a composed function you must also consider f(x),

So the Domain is all non-negative Real Numbers

Why Both Domains?

Well, imagine the functions were machines ... the first one melts a hole with a flame (only for metal), the second one drills the hole a little bigger (works on wood or metal):

What you see at the end is a drilled hole, and you may think "that should work for wood or metal". But if you put wood into g º f then the first function f would make a fire and burn everything down!

So what happens "inside the machine" is important.

De-Composing Function

You can go the other way and break up a function into a composition of other functions.

Example: (x+1/x)²

That function could have been made from these two functions:

f(x) = x + 1/x

g(x) = x²

And we would have:

(g º f)(x) = g(f(x)) = g(x + 1/x) = (x + 1/x)²

This can be useful if the original function is too complicated to work on.

Summary

"Function Composition" is applying one function to the results of another.
(g º f)(x) = g(f(x)), first apply f(), then apply g()
You must also respect the domain of the first function
Some functions can be de-composed into two (or more) simpler functions.