# 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) = x**^{2}

^{2}

**"x" is just a placeholder**. To avoid confusion let's just call it "input":

**f(input) = 2(input)+3**

**g(input) = (input) ^{2}**

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) = 2x^{2}+3

**We get a different result!**

When we reverse the order the result is rarely the same.

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 means **multiply**.

## Composed With Itself

We 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

We should be able to do it without the pretty diagram:

## Domains

It has been easy so far, but now we 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 we give it, so it is **up to us** to make sure we get the domain correct!

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

We can't have the square root of a negative number (unless we 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 we will get bad results!

## Domain of Composite Function

We 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 we 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) = x**^{2}

^{2}

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

The Domain of **g(x) = x ^{2}** is all the Real Numbers

The composed function is:

^{2}

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

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

So the Domain is all non-negative Real Numbers

## Why Both Domains?

Well, imagine the functions are 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 we see at the end is a drilled hole, and we may think "that should work for wood But if we put wood into g º f then the first function f will make a fire and burn everything down! |

So what happens "inside the machine" is important.

## De-Composing Function

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

### Example: ** (x+1/x)**^{2}

^{2}

That function can be made from these two functions:

f(x) = x + 1/x

g(x) = x^{2}

And we get:

^{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()- We must also respect the domain of the first function
- Some functions can be de-composed into two (or more) simpler functions.