Which of the following functions returns the largest integer that is less than or equal to x

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The greatest Integer Function [X] indicates an integral part of the real number 

In general: If, 

In the above figure, we are taking the floor of the values each time. When the intervals are in the form of [n, n+1), the value of the greatest integer function is n, where n is an integer.  

1. 0<=x<1 will always lie in the interval [0, 0.9), so here the Greatest Integer Function of X will be 0.
2. 1<=x<2 will always lie in the interval [1, 1.9), so here the Greatest Integer Function of X will be 1.
3. 2<=x<3 will always lie in the interval [2, 2.9), so here the Greatest Integer Function of X will be 2.

Examples:  

Number Line Representation

• If we examine a number line with the integers and plot 2.7 on it, we see: 
  • The largest integer that is less than 2.7 is 2. So [2.7] = 2. 
  • If we examine a number line with the integers and plot -1.3 on it, we see: 

Note: In the above graph, the left endpoint at 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 part of the graph.

Properties of Greatest Integer Function: 

• [X]=X holds if X is an integer.
• [X+I]=[X]+I, if I is an integer, then we can I separately in the Greatest Integer Function.
• [X+Y]>=[X]+[Y], means the greatest integer of the sum of X and Y is the equal sum of the GIF of X and the GIF of Y.
• If [f(X)]>=I, then f(X) >= I.
• If [f(X)]<=I, then f(X) < I+1.
• [-X]= -[X], If X
  Integer.
• [-X]=-[X]-1, If X is not an Integer.

It is also known as the stepwise function or floor of X.

The below program shows the implementation of the Greatest Integer Function using floor() method. 

C++

#include

using namespace std;

int GIF(float n)

{

    return floor(n);

}

int main()

{

    int n = 2.3;

    cout << GIF(n);

    return 0;

}

Java

class GFG{

static int GIF(double n)

{

    return (int)Math.floor(n);

}

public static void main(String[] args)

{

    double n = 2.3;

    System.out.println(GIF(n));

}

}

Python3

import math

def GIF(n):

    return int(math.floor(n));

n = 2.3;

print(GIF(n));

C#

using System;

class GFG{

static int GIF(double n)

{

    return (int)Math.Floor(n);

}

static void Main()

{

    double n = 2.3;

    Console.WriteLine(GIF(n));

}

}

PHP

function GIF($n)

{

    return floor($n);

}

    $n = 2.3;

    echo GIF($n);

?>

Javascript

Time Complexity: O(1)

Auxiliary Space: O(1)