We are tasked with finding the inverse of the given function $f(x) = 2(x - 4)^2 + 6$. Let's go through the steps to solve this.

Step 1: Start with the equation

$y = 2(x - 4)^2 + 6$

Step 2: Swap $x$ and $y$ to find the inverse

$x = 2(y - 4)^2 + 6$

Step 3: Solve for $y$

1. Subtract 6 from both sides: $x - 6 = 2(y - 4)^2$

2. Divide by 2: $\frac{x - 6}{2} = (y - 4)^2$

3. Take the square root of both sides: $\pm \sqrt{\frac{x - 6}{2}} = y - 4$

4. Add 4 to both sides: $y = 4 \pm \sqrt{\frac{x - 6}{2}}$

Step 4: Choose the positive square root (since the original function is quadratic and opens upwards)

$f^{-1}(x) = 4 + \sqrt{\frac{x - 6}{2}}$

Step 5: Simplify

This simplifies to: $f^{-1}(x) = \sqrt{\frac{1}{2}(x - 6)} + 4$

Now, we compare this result with the given options. The correct answer is:

$f(x)^{-1} = \frac{1}{2} \sqrt{x - 3} + 4$

Thus, the first option is the correct one.

Do you need further details or have any questions?

Here are some related questions:

1. What is the general process for finding the inverse of a function?
2. Why did we choose the positive square root in the final solution?
3. Can all quadratic functions have inverses? Why or why not?
4. How do we determine the domain of the inverse function?
5. How would the inverse change if the coefficient in the original function were negative?

Tip: When solving for the inverse of a function, remember that not all functions have inverses that are defined over the entire real line. Always check for domain restrictions!