I see that you've uploaded a file that contains a set of mathematics questions. Based on the content, it appears to involve functions, including finding inverses, completing squares, determining domains and ranges, graphing, and understanding transformations.

Here’s a breakdown of the steps based on your request:

1. State the domain of the function in its natural state without the restriction on the domain: The function $f(x) = x^2 - 6x - 16$ is a quadratic function, and its domain without restriction is all real numbers, i.e., $(-\infty, \infty)$.

2. Why is it necessary to restrict the domain when finding the inverse?: Quadratic functions are not one-to-one, meaning they fail the horizontal line test, so their inverses cannot be obtained unless we restrict the domain. Typically, we restrict the domain to $x \geq 3$ (since that part of the parabola is increasing, making the function one-to-one).

3. Rewriting the function by completing the square: To complete the square for the function $f(x) = x^2 - 6x - 16$, we can follow these steps:

   • Start by factoring the quadratic expression $x^2 - 6x$.
   • To complete the square, take half of $-6$ (which is $-3$), and square it to get $9$.
   • Rewrite the expression as $f(x) = (x^2 - 6x + 9) - 9 - 16 = (x - 3)^2 - 25$.

4. Finding the inverse function: The inverse of the function $f(x) = (x - 3)^2 - 25$ involves swapping $x$ and $y$ and solving for $y$:

   • Start with $x = (y - 3)^2 - 25$.
   • Add 25 to both sides: $x + 25 = (y - 3)^2$.
   • Take the square root of both sides: $y - 3 = \pm \sqrt{x + 25}$.
   • Since we restricted the domain to $x \geq 3$, we take the positive root: $y = 3 + \sqrt{x + 25}$.

5. Domain and range of $f(x)$:

   • Domain of $f(x)$ (with restriction $x \geq 3$): $[3, \infty)$.
   • Range of $f(x)$ (since the function opens upwards): $[-25, \infty)$.

Would you like me to assist with other specific parts of this set of questions or explain any concepts in more detail?