The greatest integer function, also known as the floor function, is a mathematical function that maps a real number to the largest integer less than or equal to that number. It is denoted as ⌊x⌋, where x is the input value. For example, ⌊3.7⌋ = 3 and ⌊-2.3⌋ = -3.

The graph of the greatest integer function consists of a series of horizontal line segments that jump at each integer value. It remains constant between integers and drops down at each integer.

The domain of the greatest integer function is all real numbers, denoted as (-∞, ∞). The range of the function is all integers, denoted as {..., -3, -2, -1, 0, 1, 2, 3, ...}.

The greatest integer function, often referred to as the floor function, is a useful mathematical tool that helps us understand how real numbers relate to integers. This function is represented as ⌊x⌋, where x is any real number. Its primary role is to return the largest integer that is less than or equal to x.

Examples of the Greatest Integer Function

To illustrate how this function works, consider the following examples:

• For the value ⌊3.7⌋, the output is 3.
• For the value ⌊-2.3⌋, the output is -3.

Graphical Representation

The graph of the greatest integer function features a series of horizontal line segments. These segments remain constant between integer values and drop down at each integer point. This creates a step-like appearance, where the function "jumps" at every integer.

Domain and Range

The domain of the greatest integer function includes all real numbers, which can be expressed as (-∞, ∞). In contrast, the range consists solely of integers, represented as {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Understanding the greatest integer function is essential for various mathematical applications, including calculus and number theory.