Given the sides $a+9$ and $a$ of a right triangle, we want to express the hypotenuse $c$ as a function of $a$.

Using the Pythagorean theorem:

$c = \sqrt{(a+9)^2 + a^2}$

Expanding the squares:

$(a+9)^2 = a^2 + 18a + 81$ $a^2 = a^2$

Adding these together:

$c = \sqrt{a^2 + 18a + 81 + a^2} = \sqrt{2a^2 + 18a + 81}$

So, the hypotenuse $c$ as a function of $a$ is:

$c(a) = \sqrt{2a^2 + 18a + 81}$

Would you like more details or have any questions?

Here are five related questions:

1. How would the function change if one of the sides was $a-9$ instead of $a+9$?
2. Can you find the derivative of the hypotenuse function $c(a)$ with respect to $a$?
3. What is the value of the hypotenuse when $a = 5$?
4. How would you solve for $a$ if the hypotenuse $c$ is known?
5. How does the hypotenuse change as $a$ increases?

Tip: When simplifying square roots, always check if the expression can be factored to make the process easier.