Greatest Integer Function: Overview, Questions, Preparation

The greatest integer function, denoted by [x], is any real function that rounds off a real number down to an integer less than that number. We also call this function a ‘Floor Function.’

For example, consider the number 1.15. According to the greatest integer function,
[1.15] = 1.
Similarly, the greatest integer of 4.56 is [4.56] = 4.

Domain and Range

We can represent the domain of the greatest integer function as (x, x+1), where ‘x’ is an integer.

So, the domain of the greatest integer function is (-4, 3), (-3, 2), (-2, 1), and so on.

This domain is the set of all real numbers. Similarly, its range is the set of all integers.

Graph of the Greatest Integer Function

We also call the greatest integer function a ‘step function.’ So, we can draw a graph for the greatest integer function, f (x) = [x].

This topic is a part of the chapter ‘Relations and Functions’ prescribed for class 12. Here, you will learn about not only the greatest integer function and its applications but also about its domain, range, and its graph. Note that the chapter carries ten marks in the annual examination.

1. Find the greatest integer function for the values [2], [1.32], [-14.982], [3.98792], [- 8.20322], and [140].

Solution.

[-2] lies between the points (-1, -3). So, the greatest integer function, in this case, would be–2.

Similarly, [1.32] lies between 1 and 2. So the greatest integer function would be 1.

The third number is [- 14.982]. And this number lies between - 14 and - 15. So, the greatest integer function, in this case, would be the larger number, - 15.

The next number, [3.98792} lies between ‘3’ and ‘4.’ So, when we consider the largest integer function for this number, we get–2.

Likewise, the number [- 8.20322] lies between–8 and–9. So, the largest integer function here would be 8.

And finally, [140] is a point on the number line. So, the largest integer function would be 140.

2.Prove that the f (x) = [x] where f: R → R, is neither one-to-one nor an onto function.

Solution.

Here, f (x) = [x] where f: R → R.
But if f (1.2) = [1.2] = 1 and f (1.9) = [1.9] = 1.
However, 1.2 is not equal to 1.9.
Therefore, the function is not one-to-one.

Now, if 0.7 ∈ R and f (x) = [x] is always an integer, there cannot exist any element x ∈ R such that f (x) = 0.7.
So, f(x) is not an onto function.

3. Evaluate [3.8].

Solution.

The integer 3.8 lies between 3 and 4. And the largest integer that is less than 3.8 is 3. So, [3.8] = 3.

Q: What is the Greatest Integer Function?

Q: What are the Domain and Range of the Greatest Integer Function?

Q: What is the Nature of the Greatest Integer Function?

Q: Is the Greatest Integer Function Periodic?

Q: Why is the Greatest Integer Function, also known as a ‘Step Function?’