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.