To find the inverse of the function $f(x) = 2(x - 4)^2 + 6$, follow these steps:

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

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

Step 2: Swap $x$ and $y$

To find the inverse, we switch $x$ and $y$: $x = 2(y - 4)^2 + 6$

Step 3: Solve for $y$

Subtract 6 from both sides:

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

Divide by 2:

$\frac{x - 6}{2} = (y - 4)^2$

Take the square root of both sides:

$y - 4 = \pm \sqrt{\frac{x - 6}{2}}$

Add 4 to both sides:

$y = 4 \pm \sqrt{\frac{x - 6}{2}}$

Step 4: Select the positive or negative branch

Since $f(x) = 2(x - 4)^2 + 6$ is a parabola that opens upwards, it’s a one-to-one function on the restricted domain where $x \geq 4$. Therefore, we select the positive square root to keep the function and its inverse consistent:

$y = 4 + \sqrt{\frac{x - 6}{2}}$

Step 5: Final answer for the inverse

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

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Would you like further details on this process or any specific clarifications?

Here are 5 questions to expand on this topic:

1. What are the domain and range of $f(x)$ and its inverse?
2. How would the inverse change if the parabola opened downwards?
3. How can you verify that the inverse is correct?
4. Why is the inverse a square root function?
5. How would the process change if the function had a different transformation?

Tip: To verify an inverse function, check that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.