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


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

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), the 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 given statement about f(x) and its inverse is true.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For the function f(x) = 1/x, the domain excludes zero, as division by zero is undefined. When analyzing the inverse function, it is essential to consider how the domain and range of f affect those of f^{-1}.

Counterexamples

A counterexample is a specific case that disproves a general statement. In the context of functions and their inverses, providing a counterexample involves finding a value for which the proposed relationship does not hold. This is a critical skill in calculus, as it helps clarify the conditions under which statements about functions and their inverses are valid.