\[\mathbf{1}\text{.\ }\]
\[y = - \cos x:\]
\[\textbf{а)}\ - \frac{\pi}{2} + 2\pi n < x < \frac{\pi}{2} + 2\pi n;\]
\[\textbf{б)}\ - \pi + 2\pi n \leq x \leq 2\pi n.\]
\[\mathbf{2.}\ \]
\[y = \sin\left( \frac{\pi}{3} - x \right):\]
\[- \frac{2\pi}{3} + 2\pi n < x < \frac{\pi}{3} + 2\pi n.\]
\[\mathbf{3.}\]
\[\cos x = \lg x;\]
\[\mathbf{\ }y = \cos x\ и\ y = \lg x:\]
\[\mathbf{О}твет:\ \ 3\ решения.\]
\[\mathbf{4}\text{.\ }\]
\[y = \sin^{2}x + 2\cos{2x}\]
\[y = \sin^{2}x + 2\left( 1 - 2\sin^{2}x \right)\]
\[y = 2 - 3\sin^{2}x.\]
\[0 \leq \sin^{2}x \leq 1\]
\[- 3 \leq - 3\sin^{2}x \leq 0\]
\[- 1 \leq 2 - 3\sin^{2}x \leq 2.\]
\[Ответ:\ \ E(y) = \lbrack - 1;\ 2\rbrack.\]
\[\mathbf{5.\ }\]
\[y = \frac{1}{2}\sin\left( 2x - \frac{\pi}{3} \right) + 1\]
\[D(x) = ( - \infty;\ + \infty);\]
\[- 1 \leq \sin\varphi \leq 1\]
\[- \frac{1}{2} \leq \frac{1}{2}\sin\varphi \leq \frac{1}{2}\]
\[\frac{1}{2} \leq \frac{1}{2}\sin\varphi + 1 \leq \frac{3}{2}\]
\[E(y) = \left\lbrack \frac{1}{2};\ \frac{3}{2} \right\rbrack.\]
\[Промежуток\ возрастания:\]
\[y^{'}(x) = \frac{1}{2} \bullet 2\cos\left( 2x - \frac{\pi}{3} \right) \geq 0\]
\[\cos\left( 2x - \frac{\pi}{3} \right) \geq 0\]
\[- \frac{\pi}{2} + 2\pi n \leq 2x - \frac{\pi}{3} \leq \frac{\pi}{2} + 2\pi n\]
\[- \frac{\pi}{6} + 2\pi n \leq 2x \leq \frac{5\pi}{6} + 2\pi n\]
\[- \frac{\pi}{12} + \pi n \leq x \leq \frac{5\pi}{12} + \pi n.\]