Let's analyze the problem given in the image.

Function and Inverse

You have a function $f(x) = x^2 - 331$, and Chivonne claims that with a domain restriction $x \geq 0$, the inverse function is $f^{-1}(x) = \sqrt{x + 331}$.

Steps to Check Chivonne's Claim:

1. Find the inverse of $f(x)$:

   • Start with the function $y = f(x) = x^2 - 331$.
   • To find the inverse, solve for $x$ in terms of $y$: $y = x^2 - 331$ $y + 331 = x^2$ $x = \pm \sqrt{y + 331}$
   • The inverse function should be $f^{-1}(y) = \pm \sqrt{y + 331}$.

2. Domain Restriction:

   • The domain restriction $x \geq 0$ eliminates the negative branch of the square root, leaving $f^{-1}(x) = \sqrt{x + 331}$.

Evaluate the Answer Choices:

• Expression for the Inverse: The expression given by Chivonne $f^{-1}(x) = \sqrt{x + 331}$ is correct under the domain restriction $x \geq 0$.
• Domain Restriction: The restriction $x \geq 0$ is indeed necessary to ensure that $f(x)$ is one-to-one and has an inverse that is a function (not a multivalued expression).

Conclusion:

Both the expression for $f^{-1}(x)$ and the domain restriction are correct.

Thus, the correct choice is:

• Both the expression for $f^{-1}(x)$ and the domain restriction are correct.

Do you want details or have any questions about this explanation?

Here are 5 related questions to deepen your understanding:

1. What would the inverse of $f(x) = x^2 - 331$ look like without the domain restriction?
2. Why is it important to restrict the domain when finding the inverse of a quadratic function?
3. Can the function $f(x) = x^2 - 331$ have an inverse if $x$ is allowed to be negative?
4. What is the domain of the inverse function $f^{-1}(x) = \sqrt{x + 331}$?
5. How would the inverse function change if the original function were $f(x) = -x^2 + 331$?

Tip: Always verify if the function is one-to-one within the chosen domain before finding its inverse. This ensures the inverse is also a function.