Function

Functions are classified by the type of mathematical equation which represents their relationship. Some functions are algebraic. Other functions like f(x) = sin x, deal with angles and are known as trigonometric. Still other functions have logarithmic and exponential relationships and are classified as such.

Algebraic functions are the most common type of function. These are functions that can be defined using addition, subtraction, multiplication, division, powers, and roots. For example f(x) = x + 4 is an algebraic function, as is f(x) = x/2 or f(x) = x³. Algebraic functions are called polynomial functions if the equation involves powers of x and constants. The most famous of these is the quadratic function (quadratic equation), f(x) = ax² + bx + c where a, b, and c are constant numbers.

A type of function that is especially important in geometry is the trigonometric function. Common trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent. One interesting characteristic of trigonometric functions is that they are periodic. This means there are an infinite number of values of x which correspond to the same value of the function. For the function f(x) = cos x, the x values 90° and 270° both give a value of 0, as do 90° + 360° = 450° and 270° + 360° = 630°. The value 360° is the period of the function. If p is the period, then f(x + p) = f(x) for all x.

Exponential functions can be defined by the equation f(x) = b^x, where b is any positive number except 1. The variable b is constant and known as the base. The most widely used base is an irrational number denoted by the letter e, which is approximately equal to 2.71828183. Logarithmic functions are the inverse of exponential functions. For the exponential function y = 4^x, the logarithmic function is its inverse, x = 4^y and would be denoted by y = f(x) = log₄ x. Logarithmic functions having a base of e are known as natural logarithms and use the notation f(x) = ln x.

We use functions in a wide variety of areas to describe and predict natural events. Algebraic functions are used extensively by chemists and physicists. Trigonometric functions are particularly important in architecture, astronomy, and navigation. Financial institutions use exponential and logarithmic functions. In each case, the power of the function allows us to take mathematical ideas and apply them to real world situations.