what is a function

Umm yes,

every x value only has one y value.

The graph has to pass the straight line test.  This means that any vertical line will never cut the graph more than once.

This might help you.

http://www.mathsisfun.com/sets/function.html

Sometimes restrictions a placed on relations so that they become functions.

eg 

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{y}}}^{{\mathtt{2}}} = {\mathtt{16}}$$    Is a circle, radius 4 and centre(0,0)

This IS a relation, because there is a relationship between x and y.

When x=0 y can be 4 or -4,  Since there are 2 possible values of y, this is NOT A FUNCTION.

However, if you restrict the range and say  $$0\le y \le 1$$     it then becomes a FUNCTION

NOTE: 

All functions are relations but not all relations are functions.

To make this easier to understand I will show a comparison.

All squares are rectangles but not all rectangles are squares.

All roses are flowers but not all flowers are roses.

All terriers are dogs but not all dogs are terriers.

A function is a relation  between a set of inputs and a set of permissable outputs with the property that each input is related to exactly one point