Let's analyze the problem step by step.

Given:

The function is $f(x) = x^3 - 11$, and you are asked to:

1. Find the inverse function $f^{-1}(x)$.
2. Verify that the inverse satisfies the conditions $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Part a: Find the inverse function

To find the inverse function $f^{-1}(x)$, follow these steps:

1. Start with the equation for the function: $y = x^3 - 11$
2. Swap $x$ and $y$ to reflect that we are finding the inverse: $x = y^3 - 11$
3. Solve for $y$: $x + 11 = y^3$ $y = \sqrt[3]{x + 11}$

Thus, the inverse function is: $f^{-1}(x) = \sqrt[3]{x + 11}$

This corresponds to choice C in the options.

Part b: Verify the inverse function

Now, you need to verify that the inverse function satisfies the conditions.

1. Verify $f(f^{-1}(x)) = x$:

$f(f^{-1}(x)) = f\left(\sqrt[3]{x + 11}\right)$ Substitute $\sqrt[3]{x + 11}$ into $f(x) = x^3 - 11$: [ f\left(\sqrt[3]{x + 11}\right) = \left(\sqrt[3]{x + 11}\right)^3 - 11 = (x + 11) - 11 = x ] So, $f(f^{-1}(x)) = x$, which is verified.

2. Verify $f^{-1}(f(x)) = x$:

$f^{-1}(f(x)) = f^{-1}(x^3 - 11)$ Substitute $x^3 - 11$ into $f^{-1}(x) = \sqrt[3]{x + 11}$: [ f^{-1}(x^3 - 11) = \sqrt[3]{(x^3 - 11) + 11} = \sqrt[3]{x^3} = x ] Thus, $f^{-1}(f(x)) = x$, which is also verified.

Conclusion

The inverse function $f^{-1}(x) = \sqrt[3]{x + 11}$ satisfies both conditions, confirming the correctness.

Would you like more details, or do you have any further questions?

Here are 5 related questions to expand on this concept:

1. How do you find the domain and range of inverse functions?
2. What are some real-world applications of inverse functions?
3. Can all cubic functions have an inverse function?
4. How would the process change if the original function was not one-to-one?
5. What are some graphical ways to verify that a function and its inverse are correct?

Tip: Always check whether the original function is one-to-one before finding the inverse.