Short Answer 
The greatest integer function, denoted as [x], maps real numbers to the greatest integer less than or equal to x, effectively rounding down any decimals. Key characteristics include a domain of all real numbers, a range of all integers, and a graph that is constant between integers with jumps at each integer. A transformation of this function can shift it vertically, such as F(x) = [x] – 2, which adjusts its intercepts accordingly.
Step 1: Understanding the Greatest Integer Function
The greatest integer function, denoted as [x], maps any real number x to the greatest integer less than or equal to x. This means that for every x, the function returns an integer value. The mathematical representation is:
- f(x) = [x]
- This function effectively “rounds down” any decimal or fractional part.
Step 2: Characteristics of the Function
The greatest integer function has several key characteristics that define its behavior and output. Understanding these characteristics helps in visualizing how the function works.
- The domain of the function includes all real numbers.
- The range comprises all integers.
- The graph of the function has a y-intercept at (0,0) and an x-intercept in the interval [0,1).
- Graphically, the function is constant between consecutive integers and jumps up one unit at each integer value.
Step 3: Transforming the Function
A transformation of the greatest integer function can be done by shifting it vertically. For instance, the function F(x) = [x] – 2 represents the original function shifted down by 2 units. This shift affects the intercepts and the overall position of the graph.
- The new y-intercept will now occur at (0, -2).
- Other x-intercepts will also shift accordingly, based on the downward movement.