## Task B-2.2.

Determine the natural domain of definition (domain) of the function $f(x)=\frac{\sqrt{x^{2}+x-12}}{|5+x|-2}-\frac{x^{2}-81}{\sqrt{10-2 x}}$.
## Solution.

To ensure the function is well-defined, the quantities under the square roots must be non-negative, and the values of the algebraic expressions in the denominators must not be equal to 0.

1. condition: $x^{2}+x-12 \geqslant 0$.

The solution to this inequality is $\langle-\infty,-4] \cup[3,+\infty\rangle$.

2. condition: $|5+x|-2 \neq 0$.

Thus, it must hold that $|5+x| \neq 2$, that is, $x \neq-3$ and $x \neq-7$.

3. condition: $10-2 x \geqslant 0$ and $10-2 x \neq 0$, that is, $10-2 x>0$.

The solution to this inequality is $\langle-\infty, 5\rangle$.

The final solution is the intersection of all conditions, that is, the natural domain of definition of the function $f$ is: $\langle-\infty,-7\rangle \cup\langle-7,-4] \cup[3,5\rangle$.