Greatest Integer Value

Integers consist of positive and negative integers, as well as zero. These numbers are a combination of natural and whole numbers. Integers do not include fractions and decimal numbers. The integer set can be described as {…..–3,–2,–1,0,1,2,3….} Integers are indeed a subset of whole and natural numbers. Integers are made up of natural numbers, zeros, and negative natural numbers. Integers are made up of whole numbers and negative natural numbers.

Integers are used to express circumstances that whole numbers cannot mathematically represent. Integers, for example, are used to express circumstances that whole numbers cannot mathematically represent. All positive, negative, and zero integers can be placed on the number line.

What is the greatest Integer Function?

The greatest integer function is another name for the step function. A function which er can take approximately to the nearest integer that has a low value or equal to the given value is referred to as the greatest integer function. The greatest integer function appears to have a step graph, which we will analyze in the next sections.

The domain and range of the greatest integer function are R and Z, respectively. The greatest integer function can be converted into the largest integer that has low value than or equal to the given value.

Range and the domain of the greatest integer function

The greatest integer function’s domain is a real number which is denoted by (R), and its range is integers and which is denoted by(Z).

Few examples of the greatest integer function

Few properties of greatest integer function are listed below: –

⌊x⌋ = x, x is integer

⌊x + n⌋ = ⌊x⌋ + n, where n ∈ Z

If ⌊f(x)] ≥ Y, then f(x) ≥ Y

⌊-x] = –⌊x], if x ∈ Z

⌊-x] =-⌊x] – 1, if x ∉ Z

Representation of integers of number line

A number line is a line with values at regular spacing or sections along it. A number line can be stretched in either direction indefinitely and is often displayed horizontally.

Difference between the greatest and least value of function integral

As we have seen the meaning of integers that contains positive and negative numbers including zero. In the process of learning, you come through two words i,e. greatest and least integral value. Let’s discuss the difference between the greatest and least value of function integral

1. Greatest integer function value:

A function approximately becomes to the nearest integer that is less than or equal to the certain value which is given is called the biggest integer function. The greatest integer function appears to have a step graph, which we will analyze in the next sections.

The domain and range of the greatest integer function are R and Z, respectively. As a result, the greatest integer function simply returns the largest integer that is less than or equal to the specified number.

For example f (3.4) = [3.4] = 3

2. Least integer function value:

The least integer function is the function that results anywhere at number x is the lowest integer larger than or equal to x. It is represented by the symbol x. It is also referred to as the x ceiling.

For example: f (2.365) = [2.365] = 3

How to find the greatest value of an integer?

Let’s start by figuring out how to determine the biggest integer value of a particular number. Remember to keep two guidelines in mind when seeking the biggest integer numbers.

1. If the value inside the brackets is not even an integer, the next smaller integer.

For instance, if f (x) = [-16.438], the two nearest numbers are -16 and -17. We always chose the lower integer for the largest integer values. As a result, [-16.438] = -17.

2. The previous figure is if the number within the brackets is an integer. This indicates that when we have g (x) = [39], the largest integer number is 39.

We can graph the biggest integer functions once we have mastered the art of obtaining the greatest integer values.

Crucial Things to remember

The following points are useful for the most significant aspects of the greatest integer function.

•         If x is an integer between n and n+1, then it equals n. If x is not a negative integer, then x=x.

•         The biggest integer function has R as its domain and Z as its range.

•         Because x is always bigger than (or equal to) x, the fractional portion is always non-negative. If x is an integer.

•         The fractional part function has a domain of R and a range of [0,1].

Conclusion

The greatest integer functions can assist us in locating the lower integer value that is near to a given number. The graph of the step function may be determined by determining the numbers of y at specific intervals of x. The graph of f(x)=[x] may be used to graph various functions.