A function is a relation in which no two ordered pairs have the same first element (or every X relates to exactly 1 Y).

Determining if a Relation is a Function

To determine if a relation is a function, just follow three easy steps.

Sketch the graph of the relation

Visually check to see the number of times a vertical line would cut the graph.

If the vertical line cuts at one place over the entire graph, then the relation is a function. However, if there is any place where the vertical line would cut at 2 or more places for one X value, then it is not a function.

example: (0,0), (8,9), (5,5), (4,2)

The relation is a function.

Writing the Equation

The equation for a function is often written as f(x)=equation. F(x) is read as "f of x".

More Linear Functions

Piecewise Linear Function

A piecewise linear function is a function whose graph, when sketched, appears broken or bent (it appears to be "in pieces", hence the name).
example: f(x)= x-2 if x>4 and 2x-6 if x<(or equal to)* 4
for graphing calculator: Y=(x-2)(x>4)+(2x-6)(x<(or equal to) 4)
To graph without a calculator, sketch both lines with no restrictions and erase those parts not required by the problem.

Step Function

Also called a greatest integer function, a step function is a function whose graph is made up of a series of horizontal segments that resemble stairsteps, hence the name.

*My apologies for the way <(or equal to) is written. I have been unable to locate the proper symbol (the < with a bar underneath), and the portion "or equal to" is an important part of the problem. Again, my apologies.
This information is taken from the chapter Functions 1 in the IB Mathematical Studies textbook.