Function

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Function

A function represents a mathematical relationship between two sets of real numbers. These sets of numbers are related to each other by a rule that assigns each value from one set to exactly one value in the other set. The standard notation for a function y = f (x), developed in the eighteenth century, is read ‘y equals f of x.’ Other representations of functions include graphs and tables. The types of rules that govern their relationships classify functions. Some of these classifications include algebraic, trigonometric, and logarithmic and exponential. It has been found by mathematicians and scientists alike, that these elementary functions can represent many real-world phenomena. In computer science, functions are used to represent data structures and to describe algorithms.

History of functions

The idea of a function was developed in the seventeenth century. During this time, French mathematician and philosopher Rene Descartes (1596–1650), in his book Geometry (1637), used the concept to describe many mathematical relationships. The term function was introduced by German mathematician Gottfried Wilhelm Leibniz (1646–1716) almost 50 years after the publication of Geometry. The idea of a function was further formalized by Swiss mathematician Leonhard Euler (1707–1783; pronounced “oiler”) who introduced the notation for a function, y = f (x).

Characteristics of functions

The idea of a function is very important in mathematics because it describes any situation in which one quantity depends on another. For example, the distance an object travels in four hours depends on its speed. When such relationships exist, one variable is said to be a function of the other. Therefore, height is a function of age and distance is a function of speed.

The relationship between the two sets of numbers of a function can be represented by a mathematical equation. Consider the relationship of the area of a square to its sides. This relationship is expressed by the equation A = x2. Here, A, the value for the area, depends on x, the length of a side. Consequently, A is called the dependent variable and x is the independent variable. In fact, for a relationship between two variables to be called a function, every value of the independent variable must correspond to exactly one value of the dependent variable.

The previous equation mathematically describes the relationship between a side of the square and its area. In functional notation, the relationship between any square and its area could be represented by f(x) = x2, where A = f(x). To use this notation, one substitutes the value found between the parentheses into the equation. For a square with a side 4 units long, the function of the area is f(4) = 4², or 16. Using f(x) to describe the function is a matter of tradition. However, one could use almost any combination of letters to represent a function such as g(s), p(q), or even LMN(z).

The set of numbers made up of all the possible values for x is called the domain of the function. The set of numbers created by substituting every value for x into the equation is known as the range of the function. For the function of the area of a square, the domain and the range are both the set of all positive real numbers. This type of function is called a one-to-one function because for every value of x, there is one and only one value of A. Other functions are not one-to-one because there are instances when two or more independent variables correspond to the same dependent variable. An example of this type of function is f(x) = x². Here, f(2) = 4 and f(-2) = 4.

Just as one added, subtracted, multiplied, or divided real numbers to get new numbers, functions can be manipulated as such to form new functions. Consider the functions f (x) = x2 and g(x) = 4x + 2. The sum of these functions f (x) +g(x) =x2 + 4x + 2. The difference of f (x) - g(x) = x2 - 4x - 2. The product and quotient can be obtained in a similar way. A composite function is the result of another manipulation of two functions. The composite function created by our previous example is noted by f(g(x)) and equal to f(4x + 2) = (4x + 2)^2. It is important to note that this composite function is not equal to the function g(f(x)).

Functions which are one-to-one have an inverse function which will undo the operation of the original function. The function f(x) = x + 6 has an inverse function denoted as f^-1(x) = x - 6. In the original function, the value for f(5) = 5 + 6=11. The inverse function reverses the operation of the first so, f^-1(11) = 11 - 6 = 5.

In addition to a mathematical equation, graphs and tables are another way to represent a function. Since a function is made up of two sets of numbers each of which is paired with only one other number, a graph of a function can be made by plotting each pair on an X,Y coordinate system known as the Cartesian coordinate system. Graphs are helpful because they allow one to visualize the relationship between the domain and the range of the function.

Classification of functions

Functions are classified by the type of mathematical equation that 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 that 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) = bx, 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 2.7182818284. Exponential functions, along with the mathematical constant e, are used in complex equations to describe growth or decay processes.

Key Terms

Dependent variable— The variable in a function whose value depends on the value of another variable in the function.

Independent variable— The variable in a function that determines the final value of the function.

Inverse function— A function that reverses the operation of the original function.

One-to-one function— A function in which there is only one value of x for every value of y and one value of y for every x.

Range— The set containing all the values of the function.

Logarithmic functions are the inverse of exponential functions. For the exponential function y = 4x, the logarithmic function is its inverse, x = 4y 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.

Mathematicians and other scientists use functions in a wide variety of areas to describe and predict natural events. Chemists and physicists use algebraic functions extensively. 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 people to take mathematical ideas and apply them to real world situations.