When looking at differential calculus, two good ways to think about functions are via algebraic geometry and interdependent variables. Neither give the proper, abstract, definition of a function, but both give a nice way of thinking about them.
Algebraic Geometry Approach
Let us set up the plane, . We choose a distinguished point called the origin and a distinguished direction which we call ‘positive ‘. Draw a line through the origin in the direction of positive . This is the -axis. Choose a unit distance for the -direction.
Now, perpendicular to the -axis, draw a line through the origin. This is the -axis. By convention positive is anti-clockwise of positive . Choose a unit distance for the -direction.
This is the plane, :
Now points on the plane can be associated with a pair of numbers . For example, the point a distance one along the positive and five along the negative can be denoted by the coordinates (1,-5):
Similarly, I can take a pair of numbers, say (-1,3), and this corresponds to a point on the plane.
This gives a duality:
points on the plane $lates \Leftrightarrow$ pairs of numbers
Now consider the completely algebraic objects