# Introduction to Functions

## Introduction to Functions

#### Mapping

A function is similar to a number machine or formula, in that you put values in to the function and out come new values. The input is x, the output is f(x). Mapping is the way of showing the results from a function. It is basically saying if you have a value of x, what would be the value of the function f(x).

Take a function f:x → 6x + 2, (which can also be written as, f(x) = 6x + 2).

If we input x = 2, we get an output of 14.

This means that the function f maps 2 to 14, or 2 → 14.

This diagram shows how each input value of x maps to only one output value of f(x). This type of mapping is called 'one-to-one mapping'.

There are certain functions where many values of x give the same value of f(x). This is called 'many-to-one mapping'. An example that shows this is:

f:x → x2 + 3x − 2.

This function maps 0 to -2 and -3 to -2.

So, some values of f(x) come from more than one input value of x.

#### Domain and Range

We need to be able to state which values of x produce values for the function f(x) and the set of these values is called the domain.

Using the example function shown above

f:x → x2 + 3x − 2.

This function produces solutions for any value of x. This means that x can be any real number - (in Further Maths there are such things as imaginary numbers!). This means we write that for,

f:x → x2 + 3x − 2, x ∈ R

(where R is the set of real numbers).

The set of output solutions produced by the function is called the range of the function.

Taking the example above, we have already noted that its minimum is at x = -17/4

Therefore all the output values of the function are greater than or equal to -17/4, which is written as: