https://file.aiquickdraw.com/mathbot/file/1729610036778lh30209m.jpg

Let's work through the given problem step by step.

### Given:
The function is \( f(x) = \sqrt[3]{x} + 12 \).

### Part (a) Find \( f^{-1}(x) \), the inverse of the function.

To find the inverse of \( f(x) \), follow these steps:
1. Start by setting \( f(x) = y \):
   \[
   y = \sqrt[3]{x} + 12
   \]
2. Solve for \( x \) in terms of \( y \):
   - First, subtract 12 from both sides:
     \[
     y - 12 = \sqrt[3]{x}
     \]
   - Cube both sides to remove the cube root:
     \[
     (y - 12)^3 = x
     \]
3. Now, express \( x \) as a function of \( y \):
   \[
   f^{-1}(x) = (x - 12)^3
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = (x - 12)^3
\]

### Part (b) Choose the correct graph.

You need to select the graph that shows both the original function \( f(x) = \sqrt[3]{x} + 12 \) and its inverse \( f^{-1}(x) = (x - 12)^3 \). In general:
- The graph of a function and its inverse will be symmetric about the line \( y = x \).
- The original function \( f(x) = \sqrt[3]{x} + 12 \) is a shifted cube root graph, and its inverse \( f^{-1}(x) = (x - 12)^3 \) is a shifted cubic graph.

From the images:
- The original function (green curve) should resemble a cube root curve shifted up by 12 units.
- The inverse function (red curve) should look like a cubic function shifted to the right by 12 units.

Based on the symmetry and the behavior of the functions, the correct graph appears to be **Option D**.

Would you like further details or clarification?

### Follow-up Questions:
1. Why does the cube root function have a different shape than a cubic function?
2. How do you confirm if a graph represents an inverse function?
3. How do transformations like shifting affect the graph of a function?
4. Why is the line \( y = x \) significant for graphs of inverses?
5. How would you graph \( f^{-1}(x) \) if given different functions?

### Tip:
When finding an inverse function, always swap \( x \) and \( y \) and solve for \( y \), which will be your new function \( f^{-1}(x) \).

Given the function f(x) = ∛x + 12, (a) Find f^(-1)(x), (b) Graph f and f^(-1) in the same rectangular coordinate system.

Learn how to find the inverse of the function f(x) = ∛x + 12 and graph both the function and its inverse. This problem covers cubic and cube root functions and inverse relationships in a rectangular coordinate system. Suitable for high school students (Grades 9-12).

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