Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 748

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\[1)\ y = tg\left( 3x - \frac{\pi}{4} \right)\]

\[\textbf{а)}\ 3x - \frac{\pi}{4} \neq \frac{\pi}{2} + \pi n\]

\[3x \neq \frac{\pi}{2} + \frac{\pi}{4} + \pi n = \frac{3\pi}{4} + \pi n\]

\[x \neq \frac{1}{3} \bullet \left( \frac{3\pi}{4} + \pi n \right) = \frac{\pi}{4} + \frac{\text{πn}}{3}.\]

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

\[\textbf{в)}\ y(x + T) = y(x)\]

\[\text{tg}\left( 3(x + T) - \frac{\pi}{4} \right) = tg\left( 3x - \frac{\pi}{4} \right)\]

\[\text{tg}\left( 3x - \frac{\pi}{4} + 3T \right) = tg\left( 3x - \frac{\pi}{4} \right)\]

\[3T = \pi\]

\[T = \frac{\pi}{3}.\]

\[\textbf{г)}\ Ни\ четная,\ ни\ нечетная:\]

\[y( - x) = tg\left( - 3x - \frac{\pi}{4} \right) =\]

\[= - tg\left( 3x + \frac{\pi}{4} \right).\]

\[\textbf{д)}\ \text{tg}\left( 3x - \frac{\pi}{4} \right) = 0\]

\[3x - \frac{\pi}{4} = arctg\ 0 + \pi n = \pi n\]

\[3x = \frac{\pi}{4} + \pi n\]

\[x = \frac{1}{3} \bullet \left( \frac{\pi}{4} + \pi n \right) = \frac{\pi}{12} + \frac{\text{πn}}{3}.\]

\[2)\ y = ctg\left( 3\left( x + \frac{\pi}{6} \right) \right) =\]

\[= \text{ctg\ }\left( 3x + \frac{\pi}{2} \right) = - tg\ 3x;\]

\[\textbf{а)}\ 3x \neq \frac{\pi}{2} + \pi n\]

\[x \neq \frac{1}{3} \bullet \left( \frac{\pi}{2} + \pi n \right) = \frac{\pi}{6} + \frac{\text{πn}}{3}.\]

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

\[\textbf{в)}\ y(x + T) = y(x)\]

\[- tg\left( 3 \bullet (x + T) \right) = - tg\ 3x\]

\[\text{tg}(3x + 3T) = tg(3x)\]

\[3T = \pi\]

\[T = \frac{\pi}{3}.\]

\[\textbf{г)}\ Функция\ нечетная:\]

\[y( - x) = - tg( - 3x) =\]

\[= tg\ 3x = - y(x).\]

\[\textbf{д)}\ - tg\ 3x = 0\]

\[tg\ 3x = 0\]

\[3x = arctg\ 0 + \pi n = \pi n\]

\[x = \frac{\text{πn}}{3}.\]