Question

what is the domain of the function the square root of 64-x^2

Answer 1

Quantities under root sign can't be less than 0.
Set >= 0:
\[64-x^2 \ge 0 \]
\[64 ≥ x^2\]
\[64 ≥ x^2\]
\[x \le 8\]

Answer 2

sorry, there is a bit more. Because you end up with a + and - when you solve a squareroot you should instead get for the domain:
.
.
\[64 \ge x^2\]
\[x \le \left| 8 \right| \]
\[-8 \le x \le 8\]