Understanding the Domain of hx fx / gx

In the realm of mathematical analysis, the domain of a function is a fundamental concept. This article aims to clarify the domain of the function hx fx / gx. Specifically, we will explore how to determine the domain of the given function and the implications of division by zero.

Introduction

The equation hx fx / gx involves the division of two functions, fx and gx. It is crucial to understand the conditions under which such a division is valid and how the domains of fx and gx interact to determine the domain of the resulting function hx.

The Intersection Principle

The principle at play here is the intersection of the domains of fx and gx. The domain of hx is defined as the set of all values of x that belong to the intersection of the domains of fx and gx. However, it is essential to exclude any values of x for which gx 0, as division by zero is undefined.

Mathematically, we can express this as:

Domain of hx domain of fx ∩ domain of gx - {x: gx 0}

Understanding the Concept

Let's break this down step-by-step to ensure a clear understanding.

Step 1: Determine the Domains of fx and gx

First, we need to identify the domains of the individual functions fx and gx. The domain of fx is the set of all x for which fx is defined. Similarly, the domain of gx is the set of all x for which gx is defined. These domains can be intervals of the real number line or more complex sets.

Step 2: Find the Intersection

The next step is to find the intersection of the domains of fx and gx. This intersection represents the set of all x values where both fx and gx are defined simultaneously. This is the common portion of the domains.

Step 3: Exclude Values Where gx 0

Finally, from the intersection obtained in the previous step, we need to exclude any values of x for which gx 0. At these points, the function hx fx / gx would be undefined, as division by zero is not defined in mathematics.

Example

Consider the functions fx x^2 - 4 and gx x - 2.

The domain of fx is all real numbers, as x^2 - 4 is defined for all x.

The domain of gx is also all real numbers, as x - 2 is defined for all x.

The intersection of the domains is all real numbers, as both functions are defined everywhere.

However, we need to exclude the value x 2 from the intersection, as gx(2) 0. Therefore, the domain of hx is all real numbers except x 2.

In mathematical notation, we can write this as:

Domain of hx (?∞, 2) ∪ (2, ∞)

Conclusion

In conclusion, understanding the domain of hx fx / gx involves identifying the domains of fx and gx, finding their intersection, and excluding the values of x for which gx 0. This process ensures that the resulting function is well-defined and avoids the undefined case of division by zero.

By mastering this concept, one can effectively analyze and manipulate rational functions in various mathematical and scientific applications.

References

Strang, G. (1991). Calculus. Wellesley-Cambridge Press.

Kreyszig, E. (2011). Advanced Engineering Mathematics. Wiley.