Understanding the Graph of f(x) Defined at Specific Points

When dealing with functions in mathematics, it is often insightful to understand how the function behaves at various points, especially where it might not be defined. Consider a function f(x) which is not defined at x0. This unique characteristic of the function leads to interesting graphical and algebraic behaviors. Let us explore the function f(x) in detail, focusing on its domain, range, and graphical features.

Overview of the Function and Its Graph

As examined, the function f(x) is not defined at x0. When approaching x0 from the positive side, we observe a significant increase in the value of f(x). Specifically, as x gets closer and closer to 0 from the positive direction, f(x) increases without bound. For instance:

At x0.000000000001, f(x) is super large. At x0.1, f(x)10. At x0.2, f(x)5. At x0.3, f(x)3. At x0.4, f(x)2. At x0.5, f(x)2. At x0.6, f(x)1. At x0.7, f(x)1. At x0.8, f(x)1. At x0.9, f(x)1. At x1, f(x)1. At x1.000001, f(x)0, and f(x)0 for all subsequent positive values of x.

This behavior indicates that as x approaches 0 from the positive side, f(x) becomes extremely large, creating a vertical asymptote at x0. However, as soon as x exceeds 0.999999, the function rapidly decreases to 0 and remains 0 for all greater values of x.

Defining the Range and Domain

Based on the behavior of the function, we can define its range and domain more formally:

Range of f(x)

The range of f(x) is all real numbers from negative infinity to 1, inclusive, and from 0 to positive infinity, exclusive. This can be written as the union of two intervals: [ -infty, -1 cup [0, infty) ].

Domain of f(x)

The domain of f(x) is all real numbers except 0, since the function is not defined at 0. In set notation, this is written as: ( mathbb{R} - {0} ).

Graphical Representation

The graph of f(x) will visually demonstrate the asymptotic behavior near x0 and the rapid decrease to 0 for ( x > 0.999999 ). Here is an illustrative sketch of the graph:

Asymptotic Behavior: The graph will have a sharp increase towards positive infinity as ( x ) approaches 0 from the positive side. Value at x1: The graph will show that f(x)1 when x1. Behavior Near x1.000001: As soon as x becomes larger than 0.999999, the graph will drop sharply to 0 and stay there.

Conclusion

Understanding the graph and behavior of functions like f(x) is crucial for advanced mathematical analysis. The unique properties of f(x), particularly its asymptotic behavior and the sudden change in value, provide valuable insights into the function's nature and can be applied in various mathematical contexts.