Greatest integer function Definition:
What is the greatest integer function? 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. This function is also known as the Floor Function.
⌊1.15⌋ = 1
⌊4.56567⌋ = 4
⌊50⌋ = 50
⌊-3.010⌋ = -4
Greatest integer function domain and range
The greatest functions are defined piecewise Its domain is a 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 function of the interval [3,4) will be 3.
The graph is not continuous. For instance, below is the graph of the function f(x) = ⌊ x ⌋.
The above graph is viewed as a group of steps and hence the integer function is also called a Step function. The left endpoint in every step is blocked(dark dot) to show that the point is a member of the graph, and the other right endpoint (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 the 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 1: Find the greatest integer function for following
According to the greatest integer function definition
(a) ⌊-261⌋ = -261
(b) ⌊3.501⌋ = 3
(c) ⌊-1.898⌋ = -2
Example 2: Evaluate ⌊3.7⌋.
On a number line, ⌊3.7⌋ lies between 3 and 4
The largest integer which is less than 3.7 is 3.
So, ⌊3.7⌋ = 3 Answer!
To know more about greatest integer function and least integer function, you can register with BYJU’S and get access to various interactive videos to make your learning easy and interesting.