The greatest integer function is also otherwise known as the step function. A function that rounds up a number to the nearest integer less than or equal to the given number is known as the biggest integer function. In the following sections, we’ll look at the steep curve of the biggest integer function. R and Z are the Domain and Range of a Function like the Greatest Integer Function.

Therefore, the greatest integer function basically rounds to the largest integer less than or equal to the provided number.

Let’s learn more about the greatest integer function, its properties, domain and range and the steps to find it in detail.

## What is the Greatest Integer Function?

For any real function, the greatest integer function is denoted by ⌊x⌋. The function takes the real number and reduces it to an integer less than the number. The Floor Function is another name for this function.

For example:

⌊1.15⌋ = 1

⌊4.56567⌋ = 4

⌊50⌋ = 50

⌊-3.010⌋ = -4

## Greatest integer function domain and range

The domain of the greatest integer function is a set of real integers separated into intervals such as [-4, 3], [-3, 2), [-2, 1], [-1, 0], and so on. Its range will consist of the integers evaluated.

Values of x (Domain) | ⌊x⌋ (Range) |

2.3 | ⌊2.3⌋ = 2 |

3.98 | ⌊3.98⌋ = 3 |

8 | ⌊8⌋ = 8 |

## Greatest Integer Function Properties

The greatest integer function’s most important properties are as follows:

- ⌊x⌋ = x
- ⌊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

Here, x is an integer and Z is a set of numbers.

## Greatest Integer Function Graph

It’s time to learn how to represent the biggest integer function on a xy-coordinate system.

- As open and closed points, we are using intervals and endpoints.

- Each interval has a shaded point on the smaller integer and an empty dot on the larger integer.

- These will subsequently be linked together by horizontal lines.

- The same steps will be followed for each interval.

- Let’s start with graphing f(x) = [x], which we can do by first building a table of values.

x | [x] | Closed Dot | Open Dot |

[-3, -2] | -3 | (-3,-3) | (-2,-3) |

[-2, -1] | -2 | (-2,-2) | (-1,-2) |

[-1, 0] | -1 | (-1,-1) | (0,1) |

[0, 1] | (0,0) | (1,0) | |

[1, 2) | 1 | (1,1) | (2,1) |

[2, 3) | 2 | (2,2) | (3,2) |

[3, 4) | 3 | (3,3) | (4,3) |

You will get the graph below.

## How to find the greatest integer value?

Let’s begin by determining how to find a number’s largest integer value. Keep the following two rules in mind when looking for the greatest integer values:

- If the number inside the brackets is not an integer, we return the next smaller integer.

For instance, if f(x) = [-15.698], the two nearest integers are -16 and -15. We always chose the smaller integer for the largest integer values. This translates to [-15.698] = -16.

- We return the original value if the number inside the brackets is an integer.

- This indicates that if g(x) = [48], the largest integer number is also 48.

We can graph the greatest integer functions once we have mastered the art of finding the greatest integer values. This section will explain why this function is also known as a step function.

## Some Important Notes

To summarise the most crucial concepts of the greatest integer function, consider the following points.

- If x is a number between n and n+1, then it equals n. In case x is a positive integer, then ⌊x⌋=x.

- The domain and range of the biggest integer function are R and Z, respectively.

- Since x is always greater than (or equal to) x, the fractional part is always non-negative. If x is an integer, its fractional part is 0.

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

Greatest Integer Function Graph Image: https://www.storyofmathematics.com/wp-content/uploads/2020/10/greatest-integer-function-300×278.png