Greatest Integer Function

Definition of Greatest Integer Function

The greatest integer function of a real number x is represented by f(x) = [x] or |_x_|

For all real numbers x, the greatest integer function returns the largest integer less than or equal to x
In other words, the greatest integer function rounds down a real number to the nearest integer

More About Greatest Integer Function

  • Greatest integer functions are piece-wise defined.
  • The domain of the greatest integer function is the set of real numbers which is divided into a number of intervals like [-4, 3), [-3, 2), [-2, 1), [-1, 0), [0, 1), [1, 2), [2, 3), [3, 4) and so on
    Hint: [a, b) is just an interval notation which means a = x < b,="" where="" x="" is="" a="" real="" number="" in="" the="" interval="" [a,="" b).="" the="" figure="" shown="" below="" is="" an="" example="" of="" a="" statistical="" graph,="" called="" the="" bar="" graph="" that="" shows="" the="" number="" of="" people="" visited="" a="" park="" in="" different="" years="">

Video Examples: The Greatest Integer Function


    When the interval is of the form [n, n + 1), where n is an integer, the value of the greatest integer function is n. For example, the value of the greatest integer function is 4 in the interval [4, 3)
  • The graph of a greatest integer function is not continuous. For example, the following is the graph of the greatest integer function f (x) = |_x_|.
    The graph above looks like a stair case (a series of steps). So, the greatest integer function is sometimes called a step function. One endpoint in each step is closed (black dot) to indicate that the point is a part of the graph and the other endpoint is open (open circle) to indicate that the points is Not a part of the graph.
  • Observe in the graph above that in each interval, the function f(x) is constant. Within an interval, the value of the function remains constant. For example: In the interval [5, 4) the value of the function f(x) remains 5.
    In different intervals, however, the function f(x) can take different constant values
  • Greatest integer function is also called floor function.

Solved Example on Greatest Integer Function

Ques :
(a) |_-256_|
(b) |_3.506_|
(c) |_-0.7_|

Solution:

    By the definition of greatest integer function
    (a) |_-256_| = -256
    (b) |_3.506_| = 3
    (c) |_-0.7_| = -1