Greatest Integer Function

The greatest integer function, also known as the step function, rounds off to the nearest integer that may be less than or equal to the given number. It is denoted inside square brackets [x]. The value is rounded off to the nearest integer. This implies that the value will always be a whole number.

For example, the value of 3.1 will be

f(x) = f(3.1) = [3].

Value of 2.999 will be f(x) = f(2.999) = [2].

Graph of the greatest integer function

The graph of the greatest integer function is a step graph.

1) For all numbers lying in the interval [0,1), the value of f will be 0.

2) For the entire interval [1,2), f will take the value 1.

3) For the interval [−1,0), f will take the value −1 and so on.

So, for every [n, n+1], the value of the greatest integer function will be n. For example, the greatest integer function of the interval [5,6) will be 5. The value of the function stays constant within an interval.

Domain and range of the greatest integer function

The domain is the real number (R), and the range is the integers (Z).

Properties

1. ⌊x+n⌋ = ⌊x⌋+n, where, n ∈ Z

2. ⌊−x⌋ { −⌊x⌋, if x ∈ Z −⌊x−1⌋, if x ∉ Z

3. If ⌊f(x)⌋ ≥ L, then f(x)≥L

Points to remember

• If x is a number between successive integers n and n+1, then ⌊x⌋=n. If x is an integer, then ⌊x⌋=x.
• The domain of the greatest integer function is R and its range is Z.
• The fractional part will always be non-negative, as x will always be greater than (or equal to) ⌊x⌋. If x is an integer, then its fractional part will be 0.
• The domain of the fractional part function is R and its range is [0,1).

Example

Find the greatest integer function for

[-561]
[4.098]
[-3.78]

Solution

According to the definition of the greatest integer function, the value of the function must be the nearest smallest integer.
[-561]
[4]
[-4]

Applications

The greatest integer function is used most in calculating bills and analyzing costs. For instance, a customer has ordered 20 bags of rice. The cost of that rice bag depends upon its weight.

So here, cost becomes a function for the rice bag’s weight. Is the weight of all the 20 rice bags the same? Will it be exact, say 50kgs, each? No! The weight can be 49.97kgs, 50.23kgs, 50.009kgs, 49.59kgs, and so on.

However, the function will be considered as 50kgs only.

Therefore, the cost of each rice bag will be defined irrespective of whatever the weight will be. This implies, f(cost) = cost of each bag weighing 50kgs.

It will be a step function with [50] as the range. This is how the greatest integer function finds its usage in various fields of economics and science.