Решебник по алгебре 11 класс Никольский Параграф 1. Функции и их графики Задание 50

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\[\textbf{а)}\ y = sgn\ x = \left\{ \begin{matrix} 1;\ \ \ \ x > 0 \\ 0;\ \ \ \ x = 0 \\ - 1;\ x < 0 \\ \end{matrix} \right.\ ;\ \]

\[\ X = R;\]

\[y = 0\ при\ x = 0;\]

\[y < 0\ при\ x < 0;\]

\[y > 0\ при\ x > 0.\]

\[\textbf{б)}\ y = sgn\ (x^{2} - 4)\]

\[y = sgn\ x = \left\{ \begin{matrix} 1;\ \ \ \ x > 0 \\ 0;\ \ \ \ x = 0 \\ - 1;\ x < 0 \\ \end{matrix} \right.\ ;\]

\[y = sgn\ \left( x^{2} - 4 \right) =\]

\[= \left\{ \begin{matrix} 1;\ \ \ \ x^{2} - 4 > 0;x < - 2\ и\ x > 2 \\ 0;\ \ \ \ x^{2} - 4 = 0;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x = \pm 2 \\ - 1;\ x^{2} - 4 < 0;\ \ \ \ \ \ \ - 2 < x < 2 \\ \end{matrix} \right.\ \]

\[y > 0\ при\ x < - 2\ и\ x > 2;\]

\[y = 0\ при\ x = \pm 2;\]

\[y < 0\ при\ \ - 2 < x < 2.\]

\[\textbf{в)}\ y = sgn\ (\lg x)\]

\[y = sgn\ x = \left\{ \begin{matrix} 1;\ \ \ \ x > 0 \\ 0;\ \ \ \ x = 0 \\ - 1;\ x < 0 \\ \end{matrix} \right.\ ;\]

\[y = sgn\ \left( \lg x \right) =\]

\[\text{=}\left\{ \begin{matrix} 1;\ \ \ \lg x > 0;\ \ \ \ \ \ \ \ \ \ \ x > 1 \\ 0;\ \ \ \lg x = 0;\ \ \ \ \ \ \ \ \ \ \ x = 1 \\ - 1;\lg x < 0;\ \ 0 < x < 1 \\ \end{matrix} \right.\ \]

\[y > 0\ при\ x > 1;\]

\[y = 0\ при\ x = 1;\]

\[y < 0\ при\ \ 0 < x < 1.\]

\[\textbf{г)}\ y = sgn\ \left( \frac{1}{x} \right);\ \ x \neq 0\]

\[y = sgn\ x = \left\{ \begin{matrix} 1;\ \ \ \ x > 0 \\ 0;\ \ \ \ x = 0 \\ - 1;\ x < 0 \\ \end{matrix} \right.\ ;\]

\[y = sgn\ \left( \frac{1}{x} \right) =\]

\[= \left\{ \begin{matrix} 1;\ \ \ \ \ \frac{1}{x} > 0;\ \ x > 0 \\ - 1;\ \frac{1}{x} < 0;\ \ x < 0 \\ \end{matrix} \right.\ \]

\[y > 0\ при\ x > 0;\]

\[y < 0\ при\ x < 0.\]