Let f(x)=(x^2)/(x^2-1). (a) From a graph of f, explain why f...

Let $f(x)=\frac{x^{2}}{x^{2}-1}$.
(a) From a graph of $f$, explain why $f$ is not invertible. Furthermore, explain why $f$ is invertible if we restrict the domain to $[0, \infty)$.
(b) Find the inverse function for $f$ restricted to $[0, \infty)$.

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Key Concepts

Function Invertibility

A function is invertible if every output value comes from exactly one input value, which is known as a one-to-one relationship. For a function to have an inverse, it must pass the horizontal line test—meaning no horizontal line intersects the graph at more than one point. If this condition is violated, the function does not have an inverse over that domain.

Domain Restriction for Invertibility

Restricting the domain of a function can enforce the one-to-one property required for invertibility. By limiting the input values to a subset where the function is monotonic (either entirely increasing or decreasing), we eliminate duplicate y-values, ensuring that each output corresponds to only one input.

Inverse Function Derivation

Finding the inverse of a function involves solving the equation y = f(x) for x. Once the equation is rearranged to express x in terms of y, the inverse function is defined by switching the roles of x and y. This process requires careful treatment of domains to maintain the function's validity, especially when square roots or rational expressions are involved.

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