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.

Graph Shifting

If you add 1 to f(x), this will shift the graph up 1 unit. Similarly, f(x) + n shifts the graph upwards by n units.
f(x - 1), in other words replacing all the x's in the formula with (x - 1), will cause the graph to shift 1 unit to the right. So f(x - n) shifts the graph n units to the right. Similarly, f(x + n) shifts the whole graph n units to the left.

f(ax), where a is some constant number, will be the graph of f(x) but it will be squashed towards the y-axis if a > 1.

Example

If f(x) = x², then f(3x) = (3x) = 9x²
So the graph of f(3x) is going to rise steeper than the graph of f(x), because of the 9.

If the number a is less than 1, then the graph is going to be stretched (by a factor of 1/a) in the direction of the x-axis.

y = af(x) will cause the graph to be stretched in the direction of the y-axis by a factor of a.

Inverse Functions

The inverse of something has the opposite effect of that thing.

The inverse function of y = 2x is y = ½x . This is because if you multiply something by 2, then by ½, you end up with what you started with.

Revision Guides; MathsRevision.Net Home

© Matthew Pinkney 2007