MA005: Calculus I

The greatest integer function of a number $x , y = f(x) = [ x ]$, is the largest integer which is less than or equal to $x$ . The value of $[ x ]$ is always an integer and $[ x ]$ is always less than or equal to $x$. For example, $[ 3.2 ] = 3, [ 3.9 ] = 3$, and $[ 3 ] = 3$. If $x$ is positive, then $[ x ]$ truncates $x$ (drops the fractional part of $x$) to get $[ x ]$. If $x$ is negative, the situation is different: $[ –4.2 ] ≠ –4$ since $–4$ is not less than or equal to $–4.2 : [ –4.2 ] = – 5, [ –4.7 ] = –5$ and $[ –4 ] = –4$. 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 $INT(x)$.

Definition of $[ x ]$: $[ x ]$ = the largest integer which is less than or equal to $x$

= $\left\{ \begin{array}{ll} x & \mbox { if x is an integer }\\  largest \; integer \; strictly \; less \; than \; x & \mbox { if x is NOT an integer}\end{array} \right.$

The domain of The $f(x) = [ x ]$ is all real numbers. The range of$f(x) = [ x ]$ is only the integers. The graph of $y = f(x) = [ x ]$ is shown in Fig. 14. It has a jump break, a step, at each integer value of $x$, and $f(x) = [ x ]$ 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 $f(x) = INT(1 + .5 sin(x) )$.

Solution: One way to create this graph is to first graph $y = 1 + 0.5sin(x)$ , the thin curve in Fig. 15, and then apply the greatest integer function to y to get the thicker "square wave" pattern. 

Practice 8: Sketch the graph of $y = INT( x^2 )$ for $–2 ≤ x ≤ 2$.