One of the most important concepts in mathematics is that of a function. Although the topic of function can appear abstract, it is nothing more than a specific rule between two sets of mathematical objects. These sets are usually numbers, but they do not have to be restricted to such mundane entities. The sets might consist of more interesting objects, such as matrices or vectors. This notwithstanding, a function is nothing more than a rule that associates with each member of one set another member of the other set. Here we discuss this exciting concept in a little more detail so that the next time you see or hear about it, rather than avert your eyes or go cowering away in fear, you jump right in on the conversation.

Before we introduce the concept of function let us define what we mean by a set. A set is simply a well-defined collection of objects. Sets can be defined by listing their specific elements as in
S = {1, 2, 3} or by definition such as S = {2n| n is an integer} to define the infinite set of even integers. A function is simply a rule between two sets, such that this rule assigns to each element of the first set, call it set A, a unique element of the second set, call it set B. For example, let set A = {1, 2, 3} and B = {2, 4, 6}. We traditionally let the letter f stand for a function. We can define a function f from set A to B such that we associate 1 in A with 2 in B; 2 in A with 4 in B; and 3 in A with 6 in B.

If you have not realized it yet, we are doubling the elements of A. That is the function defined from set A to set B is that which multiplies each element or member in A (which is usually denoted by little “a”) by 2 to get each element “b” in set B. That is for each a in A we get b in B equal to 2a. Thus 2 = 2(1); 4 = 2(2); and 6 = 2(3). We write this function as f(a) = 2a. Remember that f(a) produces an element in the second set B.

In mathematics and particularly algebra, we often see the notation y = f(x). Here x in X is the first set, completely analogous with set A above; and y in Y is the second set, completely analogous with set B above. As you may realize now, functions between the letters x and y are common and x and y stand for the axes of the cartesian coordinate plane or x-y system. Depending on the complexity of the function or rule, the function y = f(x) can usually be graphed using graph paper and the pictorial relationship between the two sets X and Y can be seen and studied.

It is common to come across the terms “domain” and “range” when talking about functions. The first set in a functional relationship is called the domain and the second set is called the range. The notation f:X->Y is the symbolism which stands for the function from set X (the domain) to set Y (the range). The letters X and Y are often replaced by other characters, and it is not uncommon to use the letters g, H, and z for functions. As mentioned previously, X and Y are most common in functional notation because of convention in naming the axes in a coordinate plane by using these letters.

Functions occur throughout the realm of mathematics and indeed life. Specific functions model many real world phenomena and help solve many practical problems. For example, the function

s = -16t^2, between s (distance) and t (time) models the distance a body will free fall in time t.
The function P = A(1 + r)^t, between P (principal) and t (time) models the amount of principal accumulated after an initial deposit A at interest rate r.

Functions give us a unique opportunity to observe what we see around us—as in nature—and come up with some kind of rule which helps to explain what we experience. Without this utile concept, we would certainly not understand the world as well as we do. Functions hold the key. So the next time you see y = f(x) or b = f(a), remember you have entered the realm of functions. And oh what an interesting realm it is!

By: Joe Pagano