Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


If $f\left(x\right)=x^2+1$ , then $f^{-1}\left(x\right)=\frac{1}{x^2+1}$.

Key Concepts

Function and Inverse Function

A function maps each input to a single output, while an inverse function reverses this mapping. For a function f(x), its inverse f⁻¹(x) satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f⁻¹. Understanding this relationship is crucial for determining whether the proposed inverse function is correct.

One-to-One Function

A function is one-to-one (injective) if it never assigns the same value to two different domain elements. This property is essential for a function to have an inverse. If f(x) = x² + 1 is not one-to-one, it cannot have a valid inverse, which is a key consideration in evaluating the given statements.

Counterexample

A counterexample is a specific case that disproves a statement or proposition. In the context of functions, providing a counterexample involves finding an input that leads to the same output for different inputs, thereby demonstrating that the function is not one-to-one. This is a critical tool in validating or refuting claims about functions and their inverses.