Understanding the Domain and Range of the Rational Function ( f(x) frac{x}{x^2 - 1} )

Rational functions are an important class of functions in mathematics, often encountered in various fields such as physics, engineering, and economics. In this article, we will explore the domain and range of the function f(x) frac{x}{x^2 - 1}.

Domain

The domain of a function consists of all the values of x for which the function is defined. In the case of the rational function f(x) frac{x}{x^2 - 1}, the function is undefined when the denominator is zero.

Let's find when the denominator is zero:

x^2 - 1 0 implies x^2 1 implies x -1 or x 1.

Therefore, the function is undefined at x -1 and x 1. Hence, the domain of f(x) is all real numbers except x -1 and x 1:

Domain: (-∞, -1) ∪ (-1, 1) ∪ (1, ∞)

Range

To find the range, we need to analyze the behavior of the function as x approaches different values. We will consider key points and asymptotes.

Horizontal Asymptote

As x → ±∞, f(x) → 0. This indicates that y 0 is a horizontal asymptote.

Critical Points

To find critical points, we take the derivative of f(x) and set it to zero:

f'(x) frac{(x^2 - 1) - x(2x)}{(x^2 - 1)^2} frac{-x^2 - 1}{(x^2 - 1)^2} 0

This implies that the numerator is zero and thus:

-x^2 - 1 0 implies x^2 -1 which has no real solutions.

However, let's consider the behavior near the critical points and asymptotes:

- As x → -1 from the left, f(x) → -∞.

- As x → -1 from the right, f(x) → ∞.

- As x → ±∞, f(x) → 0.

By examining the function's behavior, we can conclude that the range of f(x) includes all real numbers except y 0:

Range: (-∞, 0) ∪ (0, ∞)

Conclusion

In summary, the domain and range of the function f(x) frac{x}{x^2 - 1} are as follows:

Domain: (-∞, -1) ∪ (-1, 1) ∪ (1, ∞)

Range: (-∞, 0) ∪ (0, ∞)

Common Mistakes

Not every answer posted is correct, but most of them are. It is important to carefully analyze the function and use standard mathematical techniques to determine the domain and range. Missteps can arise when not considering all critical points and potential asymptotes.

By following a structured approach and using algebraic techniques, such as discerning the behavior near asymptotes and solving for critical points, we can ensure accurate results.