0\; ,\; \; \frac{x-3}{x+5}>0\; ,\; \; +++(-5)---(3)+++\\\\x\in (-\infty ,-5)\cup (3,+\infty )\\\\3)\; \; y=ctg\frac{x}{2}\\\\OOF:\; \; \pi n<\frac{x}{2}<\pi +\pi n\; ,\; \; 2\pi n <x<2\pi +2\pi n\; ,\; n\in Z\\\\x\in (2\pi n,2\pi +2\pi n)\; ,\; n\in Z" alt="1)\; \; y=\sqrt{4-x^2}\\\\OOF:\; \; 4-x^2\geq 0\; \; ,\; \; x^2-4\leq 0\; \; ,\; \; (x-2)(x+2)\leq 0\; ,\\\\+++[-2\, ]---[\, 2\, ]+++\\\\x\in [-2,2\, ]\\\\2)\; \; y=log_7\frac{3-x}{5+x}\\\\OOF:\; \; \frac{3-x}{5+x}>0\; ,\; \; \frac{x-3}{x+5}>0\; ,\; \; +++(-5)---(3)+++\\\\x\in (-\infty ,-5)\cup (3,+\infty )\\\\3)\; \; y=ctg\frac{x}{2}\\\\OOF:\; \; \pi n<\frac{x}{2}<\pi +\pi n\; ,\; \; 2\pi n <x<2\pi +2\pi n\; ,\; n\in Z\\\\x\in (2\pi n,2\pi +2\pi n)\; ,\; n\in Z" align="absmiddle" class="latex-formula">
![4)\; \; y=arcsin\frac{1}{x+3}\\\\OOF:\; \; -1\leq \frac{1}{x+3}\leq 1\; \; ,\; \; \left \{ {{\frac{1}{x+3}\leq 1} \atop {\frac{1}{x+3}\geq -1}} \right.\; \; \left \{ {{\frac{-x-2}{x+3}\leq 0} \atop {\frac{x+4}{x+3}\geq 0}} \right. \\\\a)\; \frac{-x-2}{x+3}\leq 0\; \; ,\; \; \frac{x+2}{x+3}\geq 0\; \; ,\; \; +++(-3)---[-2\, ]+++\\\\x\in (-\infty ,-3)\cup [-2,+\infty )\\\\b)\; \frac{x+4}{x+3}\geq 0\; \; \; +++[-4\, ]---(-3)+++\\\\x\in (-\infty ,-4\, ]\cup (-3,+\infty )](https://tex.z-dn.net/?f=4%29%5C%3B%20%5C%3B%20y%3Darcsin%5Cfrac%7B1%7D%7Bx%2B3%7D%5C%5C%5C%5COOF%3A%5C%3B%20%5C%3B%20-1%5Cleq%20%5Cfrac%7B1%7D%7Bx%2B3%7D%5Cleq%201%5C%3B%20%5C%3B%20%2C%5C%3B%20%5C%3B%20%5Cleft%20%5C%7B%20%7B%7B%5Cfrac%7B1%7D%7Bx%2B3%7D%5Cleq%201%7D%20%5Catop%20%7B%5Cfrac%7B1%7D%7Bx%2B3%7D%5Cgeq%20-1%7D%7D%20%5Cright.%5C%3B%20%5C%3B%20%5Cleft%20%5C%7B%20%7B%7B%5Cfrac%7B-x-2%7D%7Bx%2B3%7D%5Cleq%200%7D%20%5Catop%20%7B%5Cfrac%7Bx%2B4%7D%7Bx%2B3%7D%5Cgeq%200%7D%7D%20%5Cright.%20%5C%5C%5C%5Ca%29%5C%3B%20%5Cfrac%7B-x-2%7D%7Bx%2B3%7D%5Cleq%200%5C%3B%20%5C%3B%20%2C%5C%3B%20%5C%3B%20%5Cfrac%7Bx%2B2%7D%7Bx%2B3%7D%5Cgeq%200%5C%3B%20%5C%3B%20%2C%5C%3B%20%5C%3B%20%2B%2B%2B%28-3%29---%5B-2%5C%2C%20%5D%2B%2B%2B%5C%5C%5C%5Cx%5Cin%20%28-%5Cinfty%20%2C-3%29%5Ccup%20%5B-2%2C%2B%5Cinfty%20%29%5C%5C%5C%5Cb%29%5C%3B%20%5Cfrac%7Bx%2B4%7D%7Bx%2B3%7D%5Cgeq%200%5C%3B%20%5C%3B%20%5C%3B%20%2B%2B%2B%5B-4%5C%2C%20%5D---%28-3%29%2B%2B%2B%5C%5C%5C%5Cx%5Cin%20%28-%5Cinfty%20%2C-4%5C%2C%20%5D%5Ccup%20%28-3%2C%2B%5Cinfty%20%29 "4)\; \; y=arcsin\frac{1}{x+3}\\\\OOF:\; \; -1\leq \frac{1}{x+3}\leq 1\; \; ,\; \; \left \{ {{\frac{1}{x+3}\leq 1} \atop {\frac{1}{x+3}\geq -1}} \right.\; \; \left \{ {{\frac{-x-2}{x+3}\leq 0} \atop {\frac{x+4}{x+3}\geq 0}} \right. \\\\a)\; \frac{-x-2}{x+3}\leq 0\; \; ,\; \; \frac{x+2}{x+3}\geq 0\; \; ,\; \; +++(-3)---[-2\, ]+++\\\\x\in (-\infty ,-3)\cup [-2,+\infty )\\\\b)\; \frac{x+4}{x+3}\geq 0\; \; \; +++[-4\, ]---(-3)+++\\\\x\in (-\infty ,-4\, ]\cup (-3,+\infty )")
![c)\; \; \left \{ {{x\in (-\infty ,-3)\cup [-2,+\infty )} \atop {x\in (-\infty ,-4\, ]\cup (-3,+\infty )}} \right. \; \; \Rightarrow \; \; \; x\in (-\infty ,-4\, ]\cup [-2,+\infty )](https://tex.z-dn.net/?f=c%29%5C%3B%20%5C%3B%20%5Cleft%20%5C%7B%20%7B%7Bx%5Cin%20%28-%5Cinfty%20%2C-3%29%5Ccup%20%5B-2%2C%2B%5Cinfty%20%29%7D%20%5Catop%20%7Bx%5Cin%20%28-%5Cinfty%20%2C-4%5C%2C%20%5D%5Ccup%20%28-3%2C%2B%5Cinfty%20%29%7D%7D%20%5Cright.%20%5C%3B%20%5C%3B%20%5CRightarrow%20%5C%3B%20%5C%3B%20%5C%3B%20x%5Cin%20%28-%5Cinfty%20%2C-4%5C%2C%20%5D%5Ccup%20%5B-2%2C%2B%5Cinfty%20%29 "c)\; \; \left \{ {{x\in (-\infty ,-3)\cup [-2,+\infty )} \atop {x\in (-\infty ,-4\, ]\cup (-3,+\infty )}} \right. \; \; \Rightarrow \; \; \; x\in (-\infty ,-4\, ]\cup [-2,+\infty )")