Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 746

\[\boxed{\mathbf{746}\mathbf{.}}\]

\[1)\ y = tg\ |x|\]

\[\textbf{а)}\ x \neq \frac{\pi}{2} + \pi n;\]

\[\textbf{б)}\ E(y) = ( - \infty;\ + \infty);\]

\[\textbf{в)}\ y(x + T) = y(x)\]

\[\text{tg}|x + T| = tg|x|\]

\[T = \pi.\]

\[\textbf{г)}\ Функция\ четная:\]

\[y( - x) = tg| - x| = tg|x| = y(x).\]

\[\textbf{д)}\ \text{tg\ }|x| = 0\]

\[|x| = arctg\ 0 + \pi n = \pi n\]

\[x = \pi n.\]

\[2)\ y = \left| \text{tg\ x} \right|\]

\[\textbf{а)}\ x \neq \frac{\pi}{2} + \pi n;\]

\[\textbf{б)}\ E(y) = \lbrack 0;\ + \infty);\]

\[\textbf{в)}\ y(x + T) = y(x)\]

\[\left| \text{tg}(x + T) \right| = \left| \text{tg\ x} \right|\]

\[\left\{ \begin{matrix} \text{tg}(x + T) = tg\ x\ \ \ \\ \text{tg}(x + T) = - tg\ x \\ \end{matrix} \right.\ \text{\ \ }\]

\[\left\{ \begin{matrix} T = \pi\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ T - не\ существует \\ \end{matrix} \right.\ \]

\[T = \pi.\]

\[\textbf{г)}\ Функция\ четная:\]

\[y( - x) = \left| \text{tg}( - x) \right| = | - tg\ x| =\]

\[= \left| \text{tg\ x} \right| = y(x).\]

\[\textbf{д)}\ \left| \text{tg\ x} \right| = 0\]

\[\text{tg\ x} = 0\]

\[x = arctg\ 0 + \pi n = \pi n.\]

\[\textbf{е)}\ Минимальные\ значения:\]

\[x = \pi n.\]

\[3)\ y = ctg\ x\]

\[\textbf{а)}\ x \neq \pi n;\]

\[\textbf{б)}\ E(y) = ( - \infty;\ + \infty);\]

\[\textbf{в)}\ y(x + T) = y(x)\]

\[\text{ctg}(x + T) = ctg(x)\]

\[T = \pi n.\]

\[\textbf{г)}\ Функция\ нечетная:\]

\[y( - x) = ctg( - x) =\]

\[= - ctg\ x = - y(x).\]

\[\textbf{д)}\ ctg\ x = 0;\]

\[x = arcctg\ 0 + \pi n = \frac{\pi}{2} + \pi n.\]

\[4)\ y = \frac{1}{\text{ctg\ x}} = tg\ x\]

\[\textbf{а)}\ x \neq \frac{\pi}{2} + \pi n;\]

\[\textbf{б)}\ E(y) = ( - \infty;\ + \infty);\]

\[\textbf{в)}\ y(x + T) = y(x)\]

\[\text{tg}(x + T) = tg(x)\]

\[T = \pi n.\]

\[\textbf{г)}\ Функция\ нечетная:\]

\[y( - x) = tg( - x) =\]

\[= - tg\ x = - y(x).\]

\[\textbf{д)}\ \text{tg\ x} = 0\]

\[x = arctg\ 0 + \pi n = \pi n.\]