Greatest Integer Function

Greatest integer function Definition:

The greatest integer function is represented/denoted by [x], for any real function. The function rounds -off the real number down to the integer less than the number.

For eg:

[1.15] = 1

[4.56567] = 4

[50] = 50

[-3.010]  = -4

Greatest integer function domain and range

The greatest functions are defined piece wise. Its domain is the group of real numbers that are divided into intervals like [-4, 3), [-3, 2), [-2, 1), [-1, 0) and so on.

Greatest integer function graph

When the intervals are in the form of (n, n+1), the value of greatest integer function is n, where n is an integer.

For example, the greatest integer funciton of the intereval [3,4) will be 3.

The graph is not continuous. For instance, below is the graph of the function f(x) = [x].

Greatest Integer Function- Graph

The above graph is viewed as a group of steps and hence the integer function is also called a Step function. The left end point in every step is blocked(dark dot) to show that the point is a member of the graph, and the other right end point (open circle) indicates the points that are not the part of the graph.

You can observe that in every interval, the function f(x) is same. The function’s value stays constant within an interval. For instance, the value of function f(x) is equal to -5 in the interval [-5, -4).

Example :

(a) [-261]

(b) [3.501]

(c) [-0.8]


According to the greatest integer function definition

(a) [-261] = -261

(b) [3.501] = 3

(c) [-1.898] = -2

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