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

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

검증된 단계별 안내

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} $.

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 $.

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.

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.

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} $.

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주요 개념

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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.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.If f(x)=1xf\(\left\)(x\(\right\))=\(\frac{1}{x}\) , then f−1(x)=1xf^{-1}\(\left\)(x\(\right\))=\(\frac{1}{x}\)