**Functions** is a topic in maths. They are a way of creating a relationship, and using it to change numbers into new numbers through that relationship.

A function has an input and an output.

## Notation[edit | edit source]

There are two ways to give functions:

f(x)=...

f:x->...

## Combination Function[edit | edit source]

Different functions can be used together to receive a different output.

You have to do the functions in a specific order. You first pass the number through the last function in the combination, and then go backwards until reaching the front. For example, if you have gf(x), you would do f first, then g.

For example:

f(x) = 4x - 3

g(x) = 4 / x

The function for g(x) may look tricky, but it's not if you do it step-by-step!

So the function gf(x) would be...

Step 1: Do the function f(x).

= f(x) = 4x - 3

Step 2: Substitute in the answer you get for Step 1 into the formula given for g(x), which here is 4 / x.

= g(4x - 3) = 4 / 4 - 3

So the answer for the composite function of gf(x) = 4 / 4 - 3!

## Inverse Functions

An inverse function is the function that if the output of the normal function goes through the input of the inverse function, you will get the number you began with.

f '(x)=...

ff '(x)=x banter

Example: g(x) = square root of (x + 1)

Find g^-1(x):

so we substitue g^-1(x) for 'y', then we want to change the subject

y = square root of (x + 1)

we need to make 1 the subject, so let's start by squaring both sides:

y^2 = x + 1

(y^2 means y squared/to the power of 2)

now to get x on its own we need to minus 1 from both sides

y^2 -1 = x

Before we write our answer, we switch the y and x around again, making it:

x^2 - 1 = y

Therefore our answer is:

g^-1(x) = x^2 - 1

100%. That's probably a 3 mark question, so well done if you understand this you got 3 marks.