Functions, Limits, and Continuity: Functions

Basic Terminology

We will frequently use the formal concept of a set, which is just a collection of objects, called elements. Examples of sets include the real numbers R, the integers, the set of names of the days in a week, and the set of letters in the alphabet. One kind of set that we will encounter fairly often is called an interval. The open interval (a, b) consists of the real numbers x such that a < x < b, while the closed interval [a, b] consists of the real numbers x such that a≤x≤b. If x is an element of the set S, we write xâààS. Thus Πâààrealnumbers, 1âàà(0, 2), and Tuesday âàà \. A function f from a set S to a set T is a rule that takes an element of the set S and gives back an element of the set T. We denote this by f : S→T. The set S is called the domain of the function f and the set T is called its range.

Suppose we have a function f : S→T, with xâààS. If f takes an element xâààS to yâààT, we write f : xy or f (x) = y, and say that "f maps x to y." We often call this element y the image of x under f, and denote it by f (x). This is illustrated in the figure below.

If f : S→T and g : T→U, then we can define a new function gof : S→U by (gof )(x) = g(f (x)) for each element xâààS. The function gof is called the composition of the functions g and f

The graph of a function is the set of all points of the form (x, f (x)). One can draw this by plotting points on a pair of coordinate axes, with the horizontal axis corresponding to x, and the vertical corresponding to f (x).

A function f : S→T is called invertible if there exists a function g : T→S such that (gof )(x) = x for each element xâààS. If f is invertible, then this function g is called the inverse of f. One way to tell if a function is invertible is to look at its graph. A function is invertible if and only if no horizontal line intersects the graph in more than one point. Take a moment to convince yourself that this is true.

Examples

(1) The most familiar functions map the set of real numbers to itself. That is, f : R→R. An example is the function f such that for each real number x, f (x) = 2x, i.e. the image of each element x is the element 2x. We may graph this function as follows: