Find The Domain Of The Function (Enter Your Answer)

Find the (implied) domain of the function. (Enter your answer using interval notation.) u(w) = w − 2 5 − w 2.

The main task is to determine the domain of the function u(w) = w - 2 / (5 - w²). The domain includes all real numbers w for which the function is defined, meaning the denominator cannot be zero because division by zero is undefined in real numbers.

Firstly, observe the function:

u(w) = (w - 2) / (5 - w²)

The denominator is 5 - w². To find where the function is undefined, set the denominator equal to zero and solve for w:

5 - w² = 0


w² = 5


w = ±√5

These are the points at which the denominator equals zero, and hence the function u(w) is undefined at w = √5 and w = -√5.

Consequently, the domain consists of all real numbers except w = √5 and w = -√5. In interval notation, this is expressed as:

Domain: (-∞, -√5) ∪ (-√5, √5) ∪ (√5, ∞)

This excludes the points where the denominator is zero, ensuring the function is defined for all other real values of w.

Paper For Above instruction

The problem involves finding the implied domain of a rational function. The given function u(w) = (w - 2) / (5 - w²) is a rational function where the domain is every real number except those that make the denominator zero. The process starts by analyzing the denominator and setting it equal to zero to find the problematic points. Solving 5 - w² = 0 yields w = ±√5 as the points where the function is undefined, as division by zero is not permitted. Therefore, the domain of u(w) is all real numbers except w = √5 and w = -√5. In interval notation, this is expressed as (-∞, -√5) ∪ (-√5, √5) ∪ (√5, ∞). This ensures the function remains well-defined and continuous over its domain, explicitly excluding points where the denominator vanishes.

Understanding the domain in this context is crucial for analyzing the behavior of rational functions and their applicable intervals. It also emphasizes the importance of identifying restrictions introduced by the denominator to accurately define the domain of such functions.

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