Understanding the Inverse Function of f(x) 2x / (x^2 - 1)

In the realm of advanced mathematics, particularly in algebra and calculus, understanding inverse functions is crucial. This article aims to delve into the inverse function of the given function f(x) 2x / (x^2 - 1) and explore the importance of domain-restriction for ensuring the function is one-to-one.

Introduction to Functions and Inverse Functions

Before we delve into the specifics, it's essential to establish a foundational understanding of what functions and their inverses are. In mathematics, a function f(x) maps elements from a set A (the domain) to another set B (the codomain). An inverse function, denoted as f-1(x), essentially undoes the operation of the original function, meaning that f(f-1(x)) x and f-1(f(x)) x as well, provided the original function is one-to-one and its range is the codomain.

Inverse of f(x) 2x / (x^2 - 1)

The given function is f(x) 2x / (x^2 - 1). To find its inverse, we must follow the algebraic process of switching x and y and solving for y. Let's begin with the equation:

y 2x / (x^2 - 1)

To find the inverse function, we need to solve for x in terms of y:

x 2y / (y^2 - 1)

After swapping x and y, the equation becomes:

y 2x / (x^2 - 1)

Solving for y, we obtain:

y -x / (xy - 2)

Thus, the inverse function is f-1(x) -x / (xy - 2).

Domain-Restriction and One-to-One Functions

It's important to note that the domain of the original function must be restricted to ensure it is one-to-one. The domain of the original function f(x) 2x / (x^2 - 1) is [-1, 1], but this interval does not make the function one-to-one without restriction.

To ensure the function is one-to-one, we must exclude x 1 and x -1 from the domain. The restricted domain for the original function is therefore (-1, 1). Consequently, the range of the original function becomes the domain of the inverse function, which is [-1, 1].

Significance of the Inverse Function

The inverse function f-1(x) -x / (xy - 2) has practical applications in various fields, including engineering, physics, and economics. It helps in solving equations, modeling relationships between variables, and facilitating the understanding of complex systems.

Conclusion

In summary, the inverse function of f(x) 2x / (x^2 - 1) is f-1(x) -x / (xy - 2), and the significance of domain-restriction cannot be overstated. By ensuring the original function is one-to-one, we can establish a clear and unambiguous relationship between the input and output, making the inverse function a powerful tool in mathematical analysis and problem-solving.