Basic Mathematical Objects

Functions

Next to sets, the most important types of object in mathematic are functions. The functions you see most often in algebra and calculus courses involve real numbers and have formulas such as x², log₂(x), and sin x. Each of these expressions specifies a real number  that is associated with a real number x; in the second case. It is necessary to say a positive real number x, since log₂(x) makes sense only if x>0. More generally, a fuction is  f, and the domain and codomain of f are A and B, respectively, the element of B that f associates with an element x of A written f(x). We Write

                                                                F: A--> B

To indicate that f is a function with domain A and codomain B. In informal terminology, we say f assigns an element of B to each element of A; or f maps an element x of A to the element f(x) of B. A slightly more preciseway of describing a function eliminates the need for such undefined  term as “assigns”,”associates”, or “map”; we shall discuss it a little later. But for most or our purposes, the informal terminology is perfectly adequate.

Sometimes there seems to be a certain arbitrariness about the codomain of a function. For example, consider the function f, whose domain is the set of real number,given by the formula f(x) = x². Let R be the set of real numbers, and let R⁺ be the set of nonnegative real number. We might say f: R à R, since for every real number x, x² is a nonnegativereal number. W might also say f : RàR⁺, since for every real number x, x² is a nonnegative real number. To confuse the matter more, if we were only  interested in considering f(x) for nonnegative value of x, we might say f : R⁺ àR, or f :R⁺ àR⁺. to be precise, these are really four different functions, even though their values for given x are all described by the same formula. To specify a function, we must say what its domain and codomain are, and we must state the rule that determines the value f(x) that assigned to each  x in the domain. This is law that is often broken : people speak of “the function f(x)=x^2”, or “ “the function log₂(x).” One common convention is to assume that the domain is the largest set for which the formula makes sence for x², R; for log₂(x), {x ͼ  R|x>0}. However, this still leaves the codomain up in the air. In many cases, it may not matter, but in defining a function we shall try to specify both sets.