Graphing the Function f(x) ex - [x]

The function f(x) ex - [x] combines the exponential growth of ex with the floor function [x], which is the integer part of x. This article will explore the key characteristics of this function, how to draw its graph, and the significance of some specific values.

Understanding the Floor Function [x]

The floor function, denoted by [x], is defined as the largest integer less than or equal to x. For example, [2.7] 2 and [-3.2] -4. When analyzing the function f(x) ex - [x], it's crucial to understand how the floor function impacts the behavior of the exponential function.

Determining the Key Points and Asymptotes

The function f(x) ex - [x] has some important points to consider:

Inflection Points: As x approaches negative infinity, the floor function [x] becomes more negative, causing the ex term to overpower it. Therefore, the inflection value of f(x) is 1/e for x → -1-, x → -2-, and so on.

Supremum Value: As x approaches 1-, 2-, and so on, the floor function [x] is just less than x, making the maximum value of f(x) e for these points.

Middle Values: For integer values of x, the function simplifies to f(x) ex - x, providing a middle value of 1 for x 0, 1, 2, etc. This is because e0 1 and e1 e, but subtracting the floor function [x] at these points results in 1.

Plotting the Graph

To plot the graph of f(x) ex - [x], follow these steps:

Identify the Inflection Points: For x values approaching negative integers by a small margin, such as -0.999 and -1.999, the function f(x) approaches 1/e.

Identify the Supremum Points: For x values just less than positive integers, such as 0.999, 1.999, and 2.999, the function f(x) approaches e.

Identify the Middle Points: For integer values, such as x 0, 1, 2, the function simplifies to f(x) ex - x, resulting in a value of 1.

Key Features and Behavior of the Function

The function f(x) ex - [x] exhibits a unique behavior due to the combination of exponential growth and the floor function:

Exponential Growth: The term ex grows very rapidly, but it is partially offset by the floor function [x], which consists of integer values.

Small x Values: For small positive x values, the function can be close to 1, as seen from f(x) e0.999 - 0.999 ≈ 1. This is because e0.999 is slightly less than e and subtracting a small positive number from e results in a value close to 1.

Large x Values: For large x values, the function resembles the exponential function ex, as the floor function [x] becomes very large but does not significantly affect the overall growth of ex.

Conclusion

By understanding the behavior of the floor function and the exponential function, we can effectively plot and analyze the function f(x) ex - [x]. The key points and values identified in this article provide a comprehensive overview of the function's graphic representation and behavior.