Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 731

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

\[\textbf{а)}\ D(x) = ( - \infty;\ + \infty);\]

\[\textbf{б)}\ - 1 \leq \sin x \leq 1\]

\[- 1 \leq \sin|x| \leq 1\]

\[E(y) = \lbrack - 1;\ 1\rbrack;\]

\[\textbf{в)}\ y(x + T) = y(x)\]

\[\sin|x + T| = \sin|x|\]

\[T = 2\pi.\]

\[\textbf{г)}\ Функция\ четная:\]

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

\[\textbf{д)}\ \sin|x| = 0\]

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

\[x = \pm \pi n = \pi n.\]

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

\[\sin|x| = 1;\]

\[|x| = \arcsin 1 + 2\pi n = \frac{\pi}{2} + 2\pi n;\]

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

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

\[\sin|x| = - 1\]

\[|x| = - \arcsin 1 + 2\pi n =\]

\[= - \frac{\pi}{2} + 2\pi n\]

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

\[2)\ y = \left| \sin x \right|\]

\[\textbf{а)}\ D(x) = ( - \infty;\ + \infty);\]

\[\textbf{б)}\ - 1 \leq \sin x \leq 1\]

\[0 \leq \left| \sin x \right| \leq 1\]

\[E(y) = \lbrack 0;\ 1\rbrack;\]

\[\textbf{в)}\ y(x + T) = y(x)\]

\[\left| \sin(x + T) \right| = \left| \sin x \right|\]

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

\[\left\{ \begin{matrix} T = 2\pi\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \sin(x + T) = \sin(x + \pi) \\ \end{matrix} \right.\ \text{\ \ }\]

\[\left\{ \begin{matrix} T = 2\pi \\ T = \pi\ \ \\ \end{matrix} \right.\ \]

\[T = \pi.\]

\[\textbf{г)}\ Функция\ четная:\]

\[y( - x) = \left| \sin( - x) \right| = \left| - \sin x \right| =\]

\[= \left| \sin x \right| = y(x).\]

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

\[\sin x = 0\]

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

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

\[\left| \sin x \right| = 1\]

\[\sin x = \pm 1\]

\[x_{1} = \arcsin 1 + 2\pi n = \frac{\pi}{2} + 2\pi n;\]

\[x_{2} = - \arcsin 1 + 2\pi n =\]

\[= - \frac{\pi}{2} + 2\pi n;\]

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

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

\[x = \pi n.\]