In the next topic, you will learn to identify the domain and range of rational functions. You will do this both graphically, by analyzing the behavior of the functions at the locations of discontinuity, and algebraically, by simplifying the terms of the function and determining when any denominator in the function equals zero, and thus the function is undefined.

The domain of a function is the collection of all possible values of the independent variable you may put into the function. For example, the domain of all polynomial functions is the entire set of real numbers; there is no real number which you put into a polynomial function which does not result in another real number.

On the other hand, the domain of a radical function is restricted. You cannot select any values for the independent variable which causes the radical term to become negative (when graphing or analyzing the radical function over the set of real numbers). Consider the function:

The domain of this function is restricted to the real numbers, x, such that x ≥ 5; any numbers less than 5 will cause the radicand to be negative, values which do not exist in the real number system.

The range of any function is the collection of all possible values of the dependent variable: the values of the function which result when you evaluate the function at one of the domain values. For example, the range of all linear functions except horizontal lines is all real numbers (for horizontal lines, of the form f(x) = k, the range is a single number, k). Yet quadratic functions have narrower ranges, as they have either a minimum or maximum value at the vertex. For example, the range of the function g(x) = x² + 3 is all real numbers y, such that y ≥ 3, since there is a minimum y value for the function at its vertex, (0,3).