Given an input, a function defines a recipe for evaluating the corresponding output. One might also call the function recipe a mapping, or a transformation, or an operator. When defining functions, we should take care to define the nature of the inputs and outputs. Mathematicians describe the input’s domain and the output’s codomain, which might be the set of real numbers, integers, complex numbers, trees, graphs, etc. Here we are concerned with scalar real variables. We must also take care to define the interval in which the function definition is valid. If there is a y which is completely determined by x in some interval a ≤ x ≤ b, then we say that y is a function of x , y = f (x), in that interval. X is the independent variable, and y the dependent variable.

Single-valued and multi-valued functions

For a single value of the independent variable x, a function might deliver one or more values of y. A single-, two-, and multi-value function.