A function is 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. For example, suppose we choose the letter x to stand for the numbers in one set and the letter y for the numbers in the second set. Then, for each value we assign to x, we can find one and only one comparable value of y.
An example of a function is the mathematical equation y = 3x + 2. For any given value of x, there is one and only one value of y. If we choose 5 for the value of x, then y must be equal to 17 (3 · 5 + 2 = 17). Or if we choose 11 for the value of x, then y must be equal to 35 (3 · 11 + 2 = 35).
The standard notation for a function is y = f(x) and is read "y equals f of x." Functions can also be represented in other ways, such as by graphs and tables. Functions are classified by the types of rules that govern their relationships: algebraic, trigonometric, logarithmic, and exponential. Mathematicians and scientists have found that elementary functions represent many real-world phenomena.
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 height of a person depends, to a certain extent, on that person's age. 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.
One way to represent the relationship between the two sets of numbers of a function is with a mathematical equation. Consider the relationship of the area of a square to its sides. This relationship is expressed by the equation A = x 2 . 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) = x 2 , where A = f(x). To use this notation, we substitute 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) = 4 2 or 16. Using f(x) to describe the function is a matter of tradition. However, we could use almost any combination of letters to represent a function such as g(s), p(q), or even LMN(z).
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.
Just as we add, subtract, multiply, or divide real numbers to get new numbers, functions can be manipulated as such to form new functions. Consider the functions f(x) = x 2 and g(x) = 4x + 2. The sum of these functions f(x) + g(x) = x 2 + 4x + 2. The difference of f(x) − g(x) = x 2 − 4x + 2. The product and quotient can be obtained in a similar way.
In addition to a mathematical equation, graphs and tables can be used 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 make it easier to visualize the relationship between the domain and the range of the function.
Functions are classified by the type of mathematical equation that represents their relationship. Algebraic functions are the most common type of function. These are functions that can be defined using addition, subtraction, multiplication, division, powers, and roots. Examples of algebraic functions include the following: f(x) = x + 4 and f(x) = x/2 and f(x) = x 3 .
Two other common types of functions are trigonometric and exponential (or logarithmic) functions. Trigonometric functions deal with the sizes of angles and include the functions known as the sine, cosine, tangent, secant, cosecant, and cotangent. 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 is known as the base.
An example of an exponential function is f(x) = 10 x . Notice that for values of x equal to 1, 2, 3, and 4, the values of f(x) are 10, 100, 1,000, and 10,000. One property of exponential functions is that they change very rapidly with changes in the independent variable.
The inverse of an exponential function is a logarithmic function. In the equation f(x) = 10 x , one procedure is to set certain values of x (as we did in the example above) and then find the corresponding values of f(x). Another possibility is to set certain values of f(x) and find out what values of x are needed to produce those values. This process is using the exponential function in reverse and is known as a logarithmic function.
All types of functions have many practical applications. Algebraic functions are used extensively by chemists and physicists. Trigonometric functions are particularly important in architecture, astronomy, and navigation. Financial institutions often use exponential and logarithmic functions.