A-level Mathematics/MEI/C3/Functions

Functions

Functions are things that take in an input and return an output. In lower years they are generally refered to as function machines. A simple function would look like this: $f(x)=2x$ . We can input a value into a function and return an answer for example substituting into out previous function the value of 2: $f(2)=2\times 2=4$. Almost all equations you meet in Core 3 will be under 4th order (i.e. no higher power than $x^{4}$ )
All functions can be talked about in terms of inputs (Domain) and co-domain or range (outputs). We often plot functions on graphs in which case the Domain goes on the x-axis and the range on the y-axis.

Combining functions

Functions can be added. Whether the 2 equations that define the function are added and then the wanted input substituted in or whether the results of the original functions are substituted in and then these answers summed, the result will always be the same.

Functions can be substituted into each other, in this case things get interesting. To illustrate this I will use two functions. I will define these 2 functions $f$ and $g$.
$f(x)=2x+1$
$g(x)=x^{2}$
$fg(x)=f(g(x))=f(x^{2})=2x^{2}+1$
$gf(x)=g(f(x))=g(2x)=(2x+1)^{2}=4x^{2}+4x+1$

As we can see from this example that equations substituted into each other do not always return the same result, in fact they very rarely do return the same result.

Mappings

There are 4 different types of mapping. These are:

Many to Many
One to Many
Many to One
One to One