Greatest Integer Function

The step function is another name for the greatest integer function. A function that rounds up a number to the nearest integer less than or equal to the given number is known as the greatest integer function. In the next sections, we will look at the steep curve of the biggest integer function. As a result, the greatest integer function simply rounds off to the largest integer that is less than or equal to the given number. We’ll learn more about the greatest integer function, its graph, and its properties in this section.

Greatest Integer Function Definition

The greatest integer less than or equal to the number is returned by the greatest integer function. x represents the greatest integer less than or equal to a number x. The specified number will be rounded to the nearest integer that is less than or equal to the number. The input variable x can obviously take on any real value. The result, however, will always be an integer. In addition, the output set will contain all integers.

Graph of Greatest Integer Function

The largest integer function graph is referred to as the step curve. This is due to the fact that the curve is structured in steps. Let’s build a graph of the integer function with the largest value. Consider the statement f(x) = x; the value of f is equal to x if and only if x is an integer. In the event that x is not a number that can be represented by an integer, the value of x will be the number that comes directly before x. 

For example,

• The value of f will be 0 for all values in the interval [0,1).

• The value of f will be 1 for the whole span [1,2).

• f will take the value 1 for the interval [1,0), and so on.

As a consequence of this, the value n will be returned by the largest integer function whenever n is an integer. The function will always return a number that is between the two numbers that are specified. The value of the function will grow by one unit when the next integer that is sent in is received. This implies that the value of f at x = 1 is really 1, and not 0, and as a result, there will be a hollow dot at (1,0) and a solid dot at (1,1), where a hollow dot indicates that the value is not included and a solid dot indicates that the value is included, respectively. 

Examples of Greatest Integer Function

Example 1: Find the greatest integer function for following

(a)  ⌊-261⌋

(b)  ⌊3.501⌋

(c) ⌊-1.898⌋

Solution:

According to the greatest integer function definition

(a) ⌊-261⌋ = -261

(b) ⌊3.501⌋ = 3

(c) ⌊-1.898⌋ = -2

Example 2:  Evaluate ⌊3.7⌋.

Solution: 

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!

Properties of Integer Function

• If x is a number that falls between n and n+1, then it is considered to be the same as n. If x is a number, then the statement x=x is validated. 

• The domain of the biggest integer function is represented by R, while its range is denoted by Z. 

• Since x is guaranteed to be higher than (or equal to) [x] at all times, the fractional component will never be in a negative value. In the event that x is a whole number, the fractional component of the value will always equal zero. 

• R serves as the fractional part function’s domain, while the interval [0,1] serves as its range. 

Conclusion

The biggest integer function, often known as the step function, is a piecewise function, and its graph looks like stairs. This leads us to the eventual conclusion that the step function is a piecewise function. The function f(x) = [x] represents the largest integer function, which may be defined as the greatest integer that is less than or equal to x. Several essential ideas connected to the Greatest Integer Function, such as its characteristics and the formulation of its graph, have been discussed in detail thus far. It is important to take note that the correct terminus of each step is shaped like an open circle.