To find the inverse of the function \( f(x) = 3(x+2)^2 - 6 \), we’ll go through the following steps:

1. **Replace \( f(x) \) with \( y \):**
   \[
   y = 3(x+2)^2 - 6
   \]

2. **Swap \( x \) and \( y \):** This step helps us solve for the inverse by switching the roles of the input and output.
   \[
   x = 3(y+2)^2 - 6
   \]

3. **Solve for \( y \):**
   - Add 6 to both sides:
     \[
     x + 6 = 3(y+2)^2
     \]
   - Divide by 3:
     \[
     \frac{x + 6}{3} = (y+2)^2
     \]
   - Take the square root of both sides. Since this is an inverse function, we consider the positive and negative roots:
     \[
     y + 2 = \pm \sqrt{\frac{x + 6}{3}}
     \]
   - Subtract 2 from both sides to isolate \( y \):
     \[
     y = -2 \pm \sqrt{\frac{x + 6}{3}}
     \]

4. **Write the inverse function:**
   \[
   f^{-1}(x) = -2 \pm \sqrt{\frac{x + 6}{3}}
   \]

The inverse is **not** a function in the strictest sense over the entire domain, as it has two branches due to the \( \pm \) symbol.

Would you like further details or have any questions?

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### Related Questions
1. How do we find the domain and range of an inverse function?
2. What makes an inverse not qualify as a function in some cases?
3. How do you restrict the domain of \( f(x) \) to ensure the inverse is a function?
4. What are some methods for verifying if two functions are inverses?
5. How does this process differ for functions that involve higher powers, such as \( x^3 \) or \( x^4 \)?

### Tip
For functions involving squares, restricting the domain to make the inverse a function is often necessary. For instance, with \( y = 3(x+2)^2 - 6 \), limiting \( x \) to \( x \geq -2 \) would provide a single branch for the inverse.

Learn how to find the inverse of the function 3(x+2)^2 - 6. This problem involves inverse functions and quadratic transformations, with a step-by-step solution that includes isolating variables and considering the square root. Ideal for advanced high school students in Grades 10-12.

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