Short Answer 
The greatest integer function, noted as [x], provides the largest integer less than or equal to x, with a domain of all real numbers and a range of integers. Its graph features discrete steps with vertical jumps at each integer, and transformations like f(x) = [x] – 3 shift the graph downward while preserving its step nature.
Step 1: Understanding the Greatest Integer Function
The greatest integer function, represented as [x], gives the largest integer that is less than or equal to x. This function is crucial in mathematics as it demonstrates how values round down to the nearest whole number. To summarize its key attributes:
- The domain includes all real numbers.
- The range consists of all integers.
- The y-intercept is located at the point (0,0).
Step 2: Analyzing the Graph Characteristics
The graph of the greatest integer function shows unique behaviors that are essential for understanding its properties. It features discrete steps between integer values, and key characteristics include:
- Constant values between consecutive integers.
- Vertical jumps of one unit at each integer.
- x-intercept falling within the interval of [0,1).
Step 3: Shifting the Function
When the function is altered by applying a transformation, such as f(x) = [x] – 3, it shifts the entire graph. This particular transformation moves the graph 3 units downward while maintaining its step characteristics. Understanding these transformations is crucial for analyzing the impacts on the function’s graph.