INSTRUCTOR: Find the domain and range of the function f of xy equals the square root of 36 minus 9x squared minus 9y squared, end of square root.

First, we want to notice we are restricted by what is inside of our radical, our radicand, and that x squared-- or minus 9 is in common between the minus 9x squared minus 9y squared. So f of xy could be written as the square root of the quantity 36 minus 9 parentheses x squared plus y squared.

Now, because x squared plus y squared is always positive-- it's greater than or equal to 0-- and because we have 36 minus 9 times that quantity, that would be undefined if x squared plus y squared was greater than 4. So 0 is less than or equal to x squared plus y squared is less than or equal to 4. That means that our domain is the set of all points xy such that x squared plus y squared is less than or equal to 4.

Now, that is the interior and boundary of a circle of a radius 2. It's the interior and the boundary of a circle with a radius equal to 2. There's our domain. Now, our range, because x squared plus y squared must be between 0 and 4, the very smallest value of the f of xy is going to be 0.

So we have bracket 0 comma. And because the largest value of x squared plus y squared can be 4-- oh, sorry, I said that backwards. If x squared plus y squared equals 0-- it's between 0 and 4-- if it's 0, then f of xy is 6. So the upper boundary on our range is 6. But if x squared plus y squared equals 4, then f of xy is 0. So our range is bracket 0 comma 6 bracket.