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

Verified step by step guidance

1. Step 1: Understand the problem statement. We are given a function \( f(x) = \frac{1}{x} \) and need to determine if its inverse \( f^{-1}(x) \) is also \( \frac{1}{x} \).
2. Step 2: Recall the definition of an inverse function. For a function \( f(x) \), its inverse \( f^{-1}(x) \) satisfies the condition \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
3. Step 3: Apply the inverse function condition to \( f(x) = \frac{1}{x} \). Assume \( f^{-1}(x) = \frac{1}{x} \) and check if \( f(f^{-1}(x)) = x \). Substitute \( f^{-1}(x) \) into \( f(x) \): \( f(f^{-1}(x)) = f\left(\frac{1}{x}\right) = \frac{1}{\left(\frac{1}{x}\right)} = x \). This condition holds.
4. Step 4: Check the second condition \( f^{-1}(f(x)) = x \). Substitute \( f(x) \) into \( f^{-1}(x) \): \( f^{-1}(f(x)) = f^{-1}\left(\frac{1}{x}\right) = \frac{1}{\left(\frac{1}{x}\right)} = x \). This condition also holds.
5. Step 5: Conclude that both conditions for inverse functions are satisfied, so the statement is true. The inverse of \( f(x) = \frac{1}{x} \) is indeed \( f^{-1}(x) = \frac{1}{x} \).

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.