To find the inverse function $f^{-1}(x)$ of the function $f(x) = x^2 - 4x + 14$, given that $x > 2$, let's proceed with the following steps:

1. Rewrite $f(x)$: $f(x) = x^2 - 4x + 14$ Since we're given that $x > 2$, we'll solve for $y$ in terms of $x$, then swap $x$ and $y$ to find the inverse function.

2. Set $y = f(x)$: $y = x^2 - 4x + 14$

3. Complete the Square to Simplify the Expression: To make it easier to solve for $x$, let's complete the square on the quadratic part of the equation: $y = (x^2 - 4x) + 14$ Completing the square for $x^2 - 4x$: $x^2 - 4x = (x - 2)^2 - 4$ Substitute this back into the equation: $y = (x - 2)^2 - 4 + 14$ Simplify: $y = (x - 2)^2 + 10$

4. Isolate $(x - 2)^2$: $y - 10 = (x - 2)^2$

5. Take the Square Root of Both Sides: Since $x > 2$, we only consider the positive square root: $x - 2 = \sqrt{y - 10}$

6. Solve for $x$: $x = 2 + \sqrt{y - 10}$

7. Swap $x$ and $y$ to Express $f^{-1}(x)$: $f^{-1}(x) = 2 + \sqrt{x - 10}$

Final Answer:

$f^{-1}(x) = 2 + \sqrt{x - 10}$

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. How would the domain of $f^{-1}(x)$ be determined based on the given $f(x)$?
2. What happens if $x \leq 2$ in the context of finding the inverse?
3. How does completing the square help in solving quadratic equations?
4. Why do we only consider the positive square root when $x > 2$?
5. Can you find the inverse of $f(x) = x^2 - 6x + 20$ for $x > 3$?

Tip: Completing the square is a powerful method to simplify quadratics and find inverse functions for quadratic expressions.