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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 77g
Chapter 1, Problem 77g

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\(\right\))=\(\frac{1}{x}\)

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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 \( \frac{1}{x} \).
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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}.
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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.
Related Practice
Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


2=ln2e2=\(\ln\)2^{e}

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Textbook Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

tan−1(tan(π4))\(\tan\)^{-1}\(\left\)(\(\tan\[\left\)(\(\frac{\pi}{4}\]\right\))\(\right\))

Textbook Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

csc−1(−1)\(\csc\)^{-1}\(\left\)(-1\(\right\))

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Textbook Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

tan−1(tan(3π4))\(\tan\)^{-1}\(\left\)(\(\tan\[\left\)(\(\frac{3\pi}{4}\]\right\))\(\right\))

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


2=10log1022=10^{\(\log\)_{10}2}

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Textbook Question

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


If f(x)=x2+1f\(\left\)(x\(\right\))=x^2+1 , then f−1(x)=1x2+1f^{-1}\(\left\)(x\(\right\))=\(\frac{1}{x^2+1}\).

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