# Functions Notation

This page aims to give an introduction to what functions in Algebra are, and how they are represented with functions notation.

### What is a Function?

A function in Algebra is a definition of a relationship between numbers.

A rule that assigns, for each number/element of a certain set, say  a  of set  A.

To a unique element of a different set, say  b  of set  B.

The situation can be illustrated to show the process.

## Functions Notation

The common notation for functions in Algebra is:    f(x)  =  y.

Where   f(x)   represents a function involving a value   x.

The letter  f  denotes the function, with an  x  value going into the function, and a  y  value coming out.

The letters for the values in function notation don’t have to always be  x  and  y.

For example,   f(a) = b   denotes a function, the value   a   goes in, and a value   b   comes out.

Also,  f  doesn’t always have to be the letter that represents the function.
We can write  h(x) = y,  which can denote a function, from  x  to  y.

Sometimes you can see a function also be denoted   f :  xy.

A function from a set  X  to a set  Y.

## Functions Domain and Range

Consider the function    f(x)  =  3x + 2.

On a set  X  comprising of the values   [ 1 , 2 , 5 , 7 ].

f(1)  =  3(1) + 2  =  3 + 2  =  5

f(2)  =  3(2) + 2  =  6 + 2  =  8

f(5)  =  3(5) + 2  =  15 + 2  =  17

f(7)  =  3(7) + 2  =  21 + 2  =  23

Set  X  is the ‘domain’, which is the set of the values that are being put into the function.

Set  Y  is the ‘range’ or image, this set contains the values that are coming out of the function.

At times set  Y  can include some extra values, that aren’t connected to any of the values in set  X  by the relevant function.
When this happens to be the case, set  Y  is classed as the as the co-domain, which we can see illustrated below.

Along with being written as   f(x)  =  3x + 2.

Such a standard function can also be written as   y  =  3x + 2.

Seeing as it’s the  y  value coming out, when an  x  value has gone in.

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