1 - Transformations and their Iteration

Summary

The concept of transformation is of fundamental importance in mathematics. Examples of transformations occur even in elementary mathematics. Functions discussed in high school text books can be regarded as transformations. For example, the quadratic polynomial f(x) = 2x² - 3x + 1 represents a transformation which carries each real number x into 2x² - 3x + 1, the latter being called the image of x under the transformation. Thus 6, 1, and 0 are the images of -1, 0, and 1, respectively. The trigonometric functions f(x) = sin x and f(x) = cos x can be regarded as transformations defined on the real axis. The logarithm function f(x) = log x is a transformation defined on the set of all positive numbers. We need to define “transformation” precisely.

Let M and S be two arbitrary sets. Suppose that a certain rule, f, associates with each element x in M a unique element y in S. Most commonly the rule is a formula for computing y from x, but y could also be obtained from x by an algorithm, a table or the observation of some quantity in the real world. The “rule” f is called a transformation from M to S, denoted by f : M → S. Furthermore we write y = f(x), and y is called the image of x under the transformation f. The terms map or function can also be used instead of transformation.