Textbook Question
Find the inverse function (on the given interval, if specified) and graph both and on the same set of axes. Check your work by looking for the required symmetry in the graphs.
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0. Functions
Common Functions
1:51 minutesProblem 77f
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
If , then .
Verified step by step guidance1
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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Video duration:
1mKey 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), 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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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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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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