Introduction to functions

Imagine a number machine: you slide a number in, it always applies the same recipe, and a new number comes out. For instance, the recipe could be: I double the number, then add 3. If you put in 5, out comes 13. In mathematics, this machine is called a function.

To talk about this machine without rewriting everything, we give it a name, often f, and we write f(x) for the number that comes out when we put x in. The recipe above is written f(x) = 2x + 3. Careful: f(x) does not mean f times x, it is a notation read as f of x.

Two vocabulary words will organise everything. The number that comes out is called the image: since f(5) = 13, we say 13 is the image of 5 under f. The number we put in is called a preimage: 5 is a preimage of 13. Remember the direction: the preimage comes before (at the input), the image comes after (at the output).

A function can also be drawn. In a coordinate system, we plot all the points with coordinates (x, f(x)): the input is read on the horizontal axis, the output on the vertical axis. All these points form the curve of the function. Reading a graph means making the trip: I start from x on the horizontal axis, I move up or down to the curve, and I read f(x) on the vertical axis.

One golden rule to finish: a number has at most one image (the machine gives only one result), but the same image can have several preimages (several inputs can give the same output). Keep this asymmetry in mind, it avoids most traps.