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GCSE Maths > Algebra - Functions

### Introduction

A function (or a 'map') is a rule which indicates an operation to perform. You can think of a function as a box which you put numbers into and get different ones out of. For example, a function might double any number which you put into it.

Functions are usually written in the form f(x) = something. The function which doubles any number you put into it is written f(x) = 2x. So if you put 3 into the function, you get 6 out (2 times 3).

e.g. if f(x) = x² + 3

then f(2) = 2² + 3 = 7   (i.e. replace x with 2)

Functions can be graphed. For example, the graph of f(x) = 1/x is as follows:

This is the same graph as y = 1/x, although the y axis is f(x) instead of y.

### Types of graphs

The graph of y = k/x (f(x) = k/x) is known as a hyperbola, where k is a constant (a fixed number). Asymptotes are lines on a graph which the graph gets very close to, but never touches. Therefore in the case of y = 1/x, the x and y axes are asymptotes.
Parabolas are graphs of the form y = ax² + bx + c (where a, b and c are numbers). They can be 'U' shaped, when a is positive, or 'n' shaped, when a is negative.

### Graph Shifting

If you add 1 to f(x), this will shift the graph up 1 unit. i.e. f(x) + n shifts the graph upwards by n units.
f(x - 1) will shift the graph 1 unit to the right. i.e. f(x - n) shifts the graph n units to the right.
f(x + n) will shift the graph n units to the left.

### Inverse Functions

The inverse function of y = 2x is y = ½x . The inverse of a function does the opposite of the function. To find the inverse of a function, follow the following procedure: let y = f(x). Swap all y's and x's . Rearrange to give y = . This is the inverse function.

#### Example

Find the inverse of f(x), where f(x) = 3x - 7
f(x) = 3x - 7
y = 3x - 7 (let f(x) = y)
x = 3y - 7 (swap x's and y's)
\ y = x + 7
3

So the inverse function is (x + 7)/3

### Combining Functions

Let f(x) = 3x + 1 and g(x) = x² + 2
Suppose we are told that f(x) + g(x) = 7 .

Then 3x + 1 + x² + 2 = 7

\x2 + 3x - 4 = 0
\ (x - 1)(x + 4) = 0
\ x = 1 or –4