A function, in Mathematics, will take a number as input and spit out an output somehow related to it. In other words, by giving a function input x, we are given y as output, having done something to x along the way in order to get this. A function can be classed as either many-to-one, in that there are a number of possible x that will give the same resultant y, or one-to-one, in that every input has its own output, and no two x can give the same y.

A function can be written as follows:

f : x --> 2x + 1   x

A function can also be written simply as f (x). A separate function of x could be written as g (x), and so on.

The above function ( f : x --> 2x + 1 ) means that for a value of x, this function will double it (2x) and then add 1 to this to give out output. So, if we were to take x = 3, we are given an output of 7. The part following this ( x ∈ ℜ ) is known as the domain, and defines which set of numbers can be used as inputs. Here, x can be any number which is a member of the set of real numbers.

A function will also have what is known as its range, which is simply the possible outputs for the function. Here, the outputs can be any real number, so the range is defined as f (x) ∈ ℜ.

If the differential of a function is always either positive or negative, then that function must be a one-to-one function. For example, imagine the function f : x |--> x 3. If we write it as y = x3, and differentiate, we are given 3x2, which will always be positive - so the function will be one-to-one.