A function is a rule that associates, with each value of a variable x in a certain set, exactly one value of another variable y. The variable y is then called the dependent variable, and x is called the independent variable. The set from which the values of x can be chosen is called the domain of the function. The set of all the corresponding values of y is called the range of the function. Examples are below:

If f(x) = x^{3} – 4x + 2, then

f(1) = 1^{3} – 4(1) + 2 = 1 – 4 + 2 = -1

f(-2) = -2^{3} – 4 (-2) + 2 = -8 + 8 + 2 = 2

f(a) = a^{3} – 4a + 2

The function f(x) = 18x – 3x^{2} is defined for every number x; that is, without exception, 18x – 3x^{2} is a real number whenever x is a real number. Thus, the domain of the function is the set of all real numbers.

The area A of a certain rectangle, one of whose sides has length x, is given by A = 18x – 3x^{2} is a real number whenever x is a real number. Thus, the domain of the function is the set of all real numbers.

The area A of a certain rectangle, one of whose sides has length x, is given by A = 18x – 3x^{2}. Here, both x and A must be positive. By completing the square, we obtain A = -3(x -3)^{2} + 27. In order to have A > 0, we must have 3(x-3)^{2} < 27. In order to have A > 0, we must have 3(x-3)^{2} < 27, which limits x to value below 6; hence, 0 < x < 6. Thus, the function determining A has the open interval (0,6) as its domain.