Understanding Domain, Range, and Inverse Asymptote of a Rational Function

When dealing with rational functions, it is important to understand the domain, range, and the asymptotes, including the inverse asymptote. This article explores these concepts in detail using a specific example, y (x^2 5x 6) / (x^3 2x^2 - x - 2). We'll also cover the step-by-step process of finding the inverse function.

Redefining the Problem

The given function is: y x^2 5x 6 / (x^3 2x^2 - x - 2)

This can be simplified as follows:

1. y (x^2 5x 6) / (x^3 2x^2 - x - 2)

2. Factorizing the denominator, we get:

x^3 2x^2 - x - 2 (x 2)(x - 1)(x 1)

3. So, our function becomes:

y (x 2)(x 3) / (x 2)(x - 1)(x 1)

4. Simplifying, we get:

y (x 3) / (x^2 - 1); for x ≠ -2

Determining the Domain

The domain of the function is the set of all real numbers for which the function is defined. In this case, the function is undefined when the denominator is zero, i.e., when:

x^2 - 1 ≠ 0

This is true for all x except -1 and 1. Additionally, the function is also undefined when x -2 due to the initial simplification. Therefore, the domain is:

Domain: x ∈ R except {-2, -1, 1}

Finding the Range

The range of a function is the set of all possible output values. To find the range, we perform a similar process to finding the domain but in the context of the inverse function. We first find the derivative of the simplified function and then determine the critical points.

The simplified function is:

y (x 3) / (x^2 - 1)

Deriving the function:

y' d/dx [(x 3) / (x^2 - 1)] -2x^2 6x 3 / (x^2 - 1)^2

Solving for critical points:

-2x^2 6x 3 0

Using the quadratic formula, we find:

x -3 ± sqrt(9 24) / -2 -3 ± sqrt(33) / 2

The values of y at these points are:

y1 (-3 sqrt(33)) / (-5) -0.086

y2 (-3 - sqrt(33)) / (-5) -2.914

Thus, the range of the function is:

Range: y ∈ R except [-2.914, -0.086]

Process to Find the Inverse Function

To find the inverse function, we first switch the x and y variables and then solve for y.

From the simplified function:

x (y 3) / (y^2 - 1)

Multiplying both sides by (y^2 - 1), we get:

x(y^2 - 1) y 3

Rearranging terms, we obtain:

xy^2 - y - 3 - x 0

This is a quadratic in y, and solving it, we get:

y 1/(2x sqrt(4x^2 12x 1))

This is the inverse function of the given rational function.

Conclusion

Understanding the domain, range, and inverse of a rational function is crucial for analyzing its behavior. By following a systematic approach, we can determine these key characteristics and understand the function's behavior more comprehensively. This process is applicable to a wide range of rational functions and is valuable for both theoretical and practical mathematical applications.