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 height of a person depends on his 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.
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, we substitute the value found between the parenthesis 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, we 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 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) = 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 you to visualize the relationship between the domain and the range of the function.