Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 1 - Functions

Problema 77f

Capitolo 1, Problema 77f

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

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

Guida verificata passo dopo passo

Step 1: Understand 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 2: Consider the given function $ f(x) = x^2 + 1 $. To find its inverse, we would typically solve the equation $ y = x^2 + 1 $ for $ x $ in terms of $ y $.

Step 3: Rearrange the equation $ y = x^2 + 1 $ to solve for $ x $: $ x^2 = y - 1 $. Then, $ x = \pm \sqrt{y - 1} $.

Step 4: Notice that the expression $ x = \pm \sqrt{y - 1} $ implies that $ f(x) = x^2 + 1 $ is not one-to-one, as it does not pass the horizontal line test. Therefore, it does not have an inverse function over the entire set of real numbers.

Step 5: The statement $ f^{-1}(x) = \frac{1}{x^2 + 1} $ is incorrect because the expression given does not satisfy the conditions for an inverse function of $ f(x) = x^2 + 1 $. A counterexample is that substituting $ x = 0 $ into $ f(x) $ gives $ f(0) = 1 $, but substituting $ x = 1 $ into $ f^{-1}(x) $ gives $ \frac{1}{2} $, which does not satisfy the inverse condition.

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Concetti chiave

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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), 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.

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Derivatives of Other Inverse Trigonometric Functions

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.

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One-Sided Limits

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.

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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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Concetti chiave

Function and Inverse Function

One-to-One Function

Counterexample

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

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