### Edexcel GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# Functions

A function is a mathematical concept that describes the relationship between two sets of numbers, known as the input and output.

In a function, each input value is associated with exactly one output value. Functions help us understand the relationship between variables.

## Function Notation

Function notation is a shorter way of expressing a function and its input-output relationship. A function is typically denoted by , where x represents the input value, and represents the corresponding output value.

For example, let’s look at the function . When the input is , the output is .

To evaluate a function for a specific input value, substitute the input value into the function rule and simplify the expression. For example, let’s evaluate the function at :

So,

## Inverse Functions

An inverse function reverses the input-output relationship of the original function. If the function f takes an input x and produces an output y, then its inverse function, denoted as , takes the input y and produces the output x.

To find the inverse of a function, follow these steps:

Step 1: Replace the function notation with the variable y.

Step 2: Swap the input and output values in the equation.

Step 3: Solve the equation for the new output variable (y).

For example, let’s find the inverse of the function :

1. Replace with :

2. Swap the input (x) and output (y) values:

3. Solve for the new output variable (y):

So, the inverse function is .

## Composite Functions

A composite function is the result of applying one function to the output of another function. If we have two functions, and , the composite function denoted as or represents applying the function g first and then applying the function f to the result.

To find a composite function, follow these steps:

Step 1: Identify the functions and .

Step 2: Write the composite function notation or .

Step 3: Substitute into the function .

For example, let’s find the composite function for and

1. Identify the functions:

2. Write the composite function notation:

3. Substitute g(x) into the function f:

So, the composite function .

## Examples

Example 1:

Given the function , find .

To find , substitute into the function:

So, .

Example 2:

Find the inverse of the function .

To find the inverse function, follow these steps:

1. Replace with :

2. Swap and :

3. Solve for :

So, the inverse function is .

Example 3:

Given the functions and , find and .

1. Find :

Substitute into the function f:

So, .

2. Find :

Substitute f(x) into the function g:

So, .

Example 4:

Given the function , find .

To find , substitute into the function:

So, .

Example 5:

Question: Find the inverse of the function .

To find the inverse function, follow these steps:

1. Replace p(x) with y:

2. Swap and :

3. Solve for y:

So, the inverse function is .

Example 6:

Question: Given the functions and , find and .

1. Find :

Substitute into the function m:

So, .

2. Find :

Substitute into the function n:

So, .