To find the inverse of the function $f(x) = 3x^2 + 6$, we need to follow these steps:

Step 1: Replace $f(x)$ with $y$

Let $y = f(x) = 3x^2 + 6$. Thus, we have: $y = 3x^2 + 6$

Step 2: Solve for $x$ in terms of $y$

To isolate $x$, we first subtract 6 from both sides: $y - 6 = 3x^2$ Now, divide both sides by 3: $\frac{y - 6}{3} = x^2$ Take the square root of both sides, remembering to account for the positive and negative square roots: $x = \pm \sqrt{\frac{y - 6}{3}}$

Step 3: Express the inverse function

Now, replace $y$ with $x$, since the inverse function will map $y$ to $x$: $f^{-1}(x) = \pm \sqrt{\frac{x - 6}{3}}$

Conclusion:

The inverse of the function $f(x) = 3x^2 + 6$ is $f^{-1}(x) = \pm \sqrt{\frac{x - 6}{3}}$.

However, because the original function $f(x) = 3x^2 + 6$ is not one-to-one (it's a parabola), we typically restrict the domain of $f(x)$ to either $x \geq 0$ or $x \leq 0$ to ensure that the inverse is a proper function. This means you could choose either the positive or negative square root depending on the domain restriction.

Would you like further details on domain restrictions or have any questions?

Here are 5 related questions:

1. How do we restrict the domain of quadratic functions to make their inverses valid?
2. What is the graphical relationship between a function and its inverse?
3. How does the process of finding inverses differ for non-quadratic functions?
4. Can we find an inverse for a function like $f(x) = 3x^2 + 6x$?
5. Why do we account for positive and negative square roots when finding inverses of quadratic functions?

Tip: Always check the domain and range of a function to ensure the inverse is valid!