Greatest Integral Value

The Greatest Integer function is basically used in mathematics to round off given real numbers. It is not continuous and not differentiable at integral points. This function is also called a step or floor function because of its graphical appearance. This article explains about greatest integer function, its domain and range, its graphical explanation, and some example problems to get a clear understanding.

Greatest Integer Function

The greatest integer function is also called the step function. It rounds off the given value to the closest integer value which is less than or equal to the given number. It is marked by the symbol [a], where a is any real number. It is a piecewise function, and its graph looks like stairs. The domain of the function is R, where R is any real number and range is Z, Z is an integer close to R. The input can be any real number, but the output will always be an integer. This function is also called as floor function.

Domain and Range

The domain of this function is R, it can take up any real number. The range of this function is Z, regardless of the input, the output is always in integers.

Value of a lies in between (-∞, +∞) and the range of the step function is always in integer.

Graph of Greatest Integer Function

The greatest integer function graph looks like a staircase, it is also called a step graph because of its step-like appearance. Let us assume a step function f(x) = [x], if x is an integer the step function will be same. If x is any non-integer, the output is an integer of lower value below x. Let n be an integer lying in between [n, n+1), then n will be the greatest integer function. This function will have a constant value anywhere between two consecutive integers. But as soon as the function meets the next integer its value increases by one unit. In numerical terms, for any value between [0, 1), the greatest integer function is 0. But when the function meets 1, its value jumps up by a unit, and the interval becomes [1,2). From the above statements, it can be concluded that for the interval [0,1) the graph will have a solid dot at (0, 0) and a hollow dot at (1, 0). As the value can never reach 1 in this interval. Whereas the graph will have a solid dot at (1, 1) and a hollow dot at (2, 1) for the interval [1, 2). Hence it is clear that the function’s domain is R, where R is any real number, and Range Z, where Z is an integer.

Properties of Greatest Integer Function

Let us look at some of the important properties of the greatest integer function,

[x] = x; if x is an integer

[x + n] = [x] + n; where n is an integer.

[-x] = – [x]; if x is an integer

[-x] = – [x] – 1; if x is not an integer.

If [f(x)] ≥ L, then f(x) ≥ L.

Examples

1.     Find the greatest integer function.
2.     [3.333]
3.     [-2.4]
4.     [√2]

In a

The given value lies in between the interval of 3 to 4, which is [3, 4). The largest integer, which is less than 3.333 is 3. So, the greatest integer function value is 3.

In b,

The given value lies in between the interval of -3 to -2, which is [-3, -2). Hence, the greatest integer value is -3.

In c,

As the value of √2 is 1.414, it lies in between the interval of 1 to 2, which is [1, 2). Hence the greatest integer value is 1.

2.     Solve the equation [3x + 4] = 11.

The given equation can be rewritten as,

11 ≤  2x + 4  < 12

Solving,

11 – 4 ≤ 2x < 12 – 4,

7 ≤ 2x < 8,

3.5 ≤ x < 4.

The equation is true for x ∈ [3.5, 4).

3.     How to represent [x] = 5 as inequalities?

The function accepts 5, and the greatest integer value of this function is 5. The interval for the function in which, any number can take up the value of 5 is [5,6). So the inequalities can be represented as 5 ≤ x < 6.

Conclusion

The Greatest Integer function helps us find the smallest integer close to the given number. In this article, we have seen what is the greatest integer function, its properties, its graphical representation, and some examples. This function’s sole purpose is to round off given values to integers.