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


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

Verified step by step guidance

1. 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 \( f^{-1}(f(x)) = x \).
2. 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 \( y \).
3. Step 3: Rearrange the equation \( y = x^2 + 1 \) to solve for \( x \): \( x^2 = y - 1 \). Then, \( x = \pm \sqrt{y - 1} \).
4. 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.
5. 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.

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), 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.

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.

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.