https://file.aiquickdraw.com/mathbot/file/17280117615270veli521.jpg

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?

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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.

The function f(x) = x^3 - 11 is one-to-one. Find the inverse f^(-1)(x) and verify that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

Learn how to find the inverse of the cubic function f(x) = x^3 - 11 and verify that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. This step-by-step solution is suitable for high school students studying algebra and functions.

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