Combinations of Functions
Greatest Integer Function: [x] or INT(x)
The greatest integer function of a number , is the largest integer which is less than or equal to . The value of is always an integer and is always less than or equal to
. For example, , and . If is positive, then truncates (drops the fractional part of ) to get . If is negative,
the situation is different: since is not less than or equal to and . On some calculators and in many programming languages the square
brackets are used for grouping objects or for lists, and the greatest integer function is represented by .
Definition of : = the largest integer which is less than or equal to
=
The domain of The is all real numbers. The range of is only the integers. The graph of is shown in Fig. 14. It has a jump break, a step, at each integer value of , and is called a step function. Between any two consecutive integers, the graph is horizontal with no breaks or holes. The greatest integer function is useful for describing phenomena which change values abruptly such as postage rates as a function of the weight of the letter ("26¢ for the first ounce and 13¢ additional for each additional half ounce"). It can also be used for functions whose graphs are "square waves" such as the on and off of a flashing light.
Example 6: Graph .
Solution: One way to create this graph is to first graph , the thin curve in Fig. 15, and then apply the greatest integer function
to y to get the thicker "square wave" pattern.