Mathematicians have a narrow technical definition of what constitutes a function. A function is a process applied to one number to get another number. For example, doubling is a function; expressed algebraically, f(n) = 2n means that the value of the function of "n" is equal to 2 times "n." Plug any number into the function, say, 7 or 13.8, and you will get a definite result, 14 or 27.6 in this case.
An important feature of functions in the standard mathematical definition is that there is exactly one result for any given input. Squaring a number is a function because for any given number n there is exactly one value for n^2. Taking the square root is not a function, however, because every positive number has two square roots, and negative numbers have only imaginary roots.
Classical geometry involves numbers, of course, but mostly hidden beneath the surface in the constructions of figures with compass and straight edge. It is in coordinate geometry, where every point on the plane is specified by a unique ordered pair (x,y), that mathematical relationships between numbers become explicit. All the classical constructions, relying on the two simple forms of circle and straight line, are reducible to sets of relationships between the x and y coordinates of points and sets of points. Yet there is an infinite variety of other relationships besides circles and lines.
In plane geometry, it is often convenient to express the y coordinate in terms of the x coordinate. The formula for a line, for example, is often written in the form y = mx + b, where "m" is a constant representing the slope of the line and "b" is the distance from the x-axis at the point where the line crosses the y-axis. You can see immediately that this form of line is a function, since every value for x. Other functional relationships include sine waves, parabolas and many other curves.
Functions that can be graphed directly on the cartesian plane are not the only functions in geometry. There are many other functional relationships, sometimes hidden in the facts about other relationships. For example, while a circle is not itself a function, since there can be two y values for each x value, the relationship between the circle's radius and its area is a function: f(r) = pi * r^2. Likewise, the total number of degrees in the angles of a polygon is also a function of the number of sides of the polygon. Identifying and understanding functional relationships like these is a primary function of mathematics.