2 - Functions

Summary

12. Functions. In elementary mathematics, it is customary to say that y is a function of x if, when x is given, y is determined (uniquely; we are not concerned with “multiple-valued functions”). This is a good working definition and one that suffices for most practical purposes. However, we should realize that it does not define “function,” although it does give a definite meaning to some phrases containing this word. (In a somewhat similar way, we are accustomed to attaching a definite meaning to the phrase “y → ∞” even though ∞ by itself has no meaning.) However, it is interesting, and sometimes helpful, actually to define a function as a genuine mathematical entity. Consider two sets E and F of real numbers, neither of which is empty, and form a class of ordered pairs (x, y) with x ∈ E and y ∈ F, where each x occurs exactly once and each y occurs at least once. Such a class of ordered pairs is called a function with domain E and range F, or a function from E to F; or, on occasions when it is unnecessary to say precisely what F is, a function with domain E and values in R¹, or a function from E into R₁, or a real-valued function with domain E, etc. For example, let E be all of R₁, let F be the closed interval [-1, 1], and let the ordered pairs be (x, sin x) for each x in R₁.