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# Introduction to Functions

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## Introduction to Functions

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 → x ^{2} + 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.

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 → x ^{2} + 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 → x ^{2} + 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: