To find the inverse of the function $f(x) = (x - 6)^2$ on the domain $[6, \infty)$, let’s go through the steps:

1. Rewrite the Function in Terms of $y$: $y = (x - 6)^2$

2. Solve for $x$ in Terms of $y$: To isolate $x$, take the square root of both sides: $\sqrt{y} = x - 6$ Since $x$ is in the domain $[6, \infty)$, we only consider the positive square root: $x = \sqrt{y} + 6$

3. Express the Inverse Function: Now, swap $x$ and $y$ to write the inverse: $f^{-1}(x) = \sqrt{x} + 6$

Final Answer:

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

Would you like a detailed explanation of each step?

Related Questions

1. How do you find the domain of an inverse function?
2. Why do we only consider the positive square root when finding the inverse here?
3. How would the inverse function change if the domain were different?
4. What does the graph of $f(x) = (x - 6)^2$ look like, and how does it relate to its inverse?
5. How can we verify that two functions are inverses of each other?

Tip

When finding the inverse of functions involving squares, always consider the domain restrictions carefully, as they determine whether to use the positive or negative root.