Greatest Integer Function: Domain, Range, Graph & Examples

Greatest Integer Function is a function that gives the greatest integer which is less than or equal to a given real number. It is a function that rounds up the number to the nearest integer less than or equal to the given number.

• Greatest Integer Function is also referred to as ‘Step Function’.
• It is denoted by the symbol ⌊x⌋, where x is any real function.
• Domain and Range of Greatest Integer Function are ℝ and ℤ respectively. 
• Examples: ⌊13.01⌋ = 13 and ⌊4.56567⌋ = 4.

Read More: NCERT Solutions For Class 11 Maths Relations and Functions

Key Terms: Greatest Integer Function, Real Number, Domain, Range, Step Function, Integer, Functions, Greatest Integer Function Graph

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What is Greatest Integer Function?

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Greatest Integer Function is defined as a function that gives the greatest integer less than or equal to a given real number. The greatest integer that is less than or equal to a number x is denoted as ⌊x⌋. 

Greatest Integer Function ⌊x⌋ is expressed mathematically as: 

The variable x can take any real value, but, the output will always be an integer.

Relations and Functions Detailed Video Explanation

Examples of Greatest Integer Function

Here are a few examples of Greatest Integer Function: 

• ⌊3.02⌋ = 3, as 3 ≤ 3.02 < 4
• ⌊50⌋ = 50
• ⌊-3.010⌋ = -4
• ⌊1.15⌋ = 1
• ⌊4.56567⌋ = 4

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Greatest Integer Function Domain and Range

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Greatest Integer Function has its domain as the set of all real numbers (ℝ), that are divided into intervals like [-4, 3), [-3, 2), [-2, 1), [-1, 0), etc. The range of the greatest integer function is the set of all integers (ℤ).

Given below are a few examples for a better understanding of the Greatest Integer Function Domain and Range:

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Greatest Integer Function Graph

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Greatest Integer Function Graph is referred to as the ‘Step Curve’ due to the step structure of the curve. In the given intervals (n, n+1), then the value of the greatest integer function is n, where n is an integer.

The graph of f(x) = ⌊x⌋ is not continuous and is represented below:

Greatest Integer Function Graph

• The graph is represented as a group of steps, thus, the Greatest Integer Function is also called Step Function.
• The left endpoint in every step is represented as a dark dot which shows that it is a part of the graph.
• The other right endpoint (open circle) is not a part of the graph.
• Another important thing that can be noticed is that in every interval, the function f(x) is the same.
• The value of the function remains constant within an interval.
• For instance, the value of function f(x) is equal to -5 in the interval [-5, -4).

Read More: Relations & Functions Important Question 

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Greatest Integer Function Properties

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Given below are the important properties of the Greatest Integer Function:

• ⌊x⌋ = x, Where x is an Integer.
• ⌊x + n⌋ = ⌊x⌋ + n, Where n ∈ Z
• ⌊-x] = –⌊x], If x ∈ Z
• ⌊-x] =-⌊x] – 1, If x ∉ Z
• If ⌊f(x)] ≥ Y, then f(x) ≥ Y
• If ⌊f(x)⌋ ≤ Y, then f(x) < Y + 1

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Greatest Integer Function Solved Examples

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Here are a few solved examples on Greatest Integer Function for a better understanding: 

Example 1: What will be the value of the Greatest Integer for the following:

1. ⌊13.01⌋
2. ⌊13.99⌋  
3. ⌊-2.4⌋

Solution: The value of Greatest Integer for the following will be

1. ⌊13.01⌋ = 13
2. ⌊13.99⌋ = 13
3. ⌊-2.4⌋ = -3

Example 2:  Determine ⌊3.7⌋.

Solution:  ⌊3.7⌋ lies between 3 and 4 on the number line. Thus, the largest integer that is less than 3.7 is 3.

Thus, ⌊3.7⌋ = 3

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Things to Remember

• Greatest Integer Function is a function that gives the greatest integer that is equal to less than the given number.
• It is denoted by the symbol ⌊x⌋, where x refers to any value.
• Mathematically, ⌊x⌋ = n, where n ≤ x < n + 1 and 'n' is an integer.
• The Domain of the Greatest Integer Function is any Real Number (ℝ).
• The Range of the Greatest Integer Function is an Integer (ℤ). 
• Greatest Integer Function Graph has a step-like curve, thus, it is also referred to as Step Curve or Step Function.

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Previous Years’ Questions

1. Let [.] denote the greatest integer function then the value of… (AIEEE - 2011)
2. Let [?] denote the greatest integer function and… (COMEDK UGET - 2011)
3. Let [.] denote the greatest integer function and f(x)... (JEE Advanced - 1993)
4. \(\displaystyle\lim_{x \to 1} [x 1]\), where [.] is the greatest integer function, is equal to…
5. If [x] denotes the greatest integer function, then… (COMEDK UGET - 2008)
6. Let [ ] denote the greatest integer function and f (x)... (VITEEE - 2008)
7. The number of discontinuity of the greatest integer function… (NTA Abhyas - 2020)
8. Here, [x] denotes the greatest integer less than or equal to… (JKCET - 2017)
9. If [x] denotes the greatest integer less than or equal to x, then the… (WBJEE - 2016)
10. The function f(x) = [x], where [x] denotes greatest integer function… (KCET - 2015)