Решебник по алгебре 11 класс Никольский Параграф 1. Функции и их графики Задание 4

\[\boxed{\mathbf{4.}}\]

\[f(x) = 7^{x};\ \varphi(x) = x^{2};\]

\[g(x) = \log_{5}x.\]

\[\textbf{а)}\ f\left( \varphi(x) \right) = 7^{\varphi(x)} = 7^{x^{2}}.\]

\[\textbf{б)}\ \varphi\left( g(x) \right) = \left( g(x) \right)^{2} = \left( \log_{5}x \right)^{2}.\]

\[\textbf{в)}\ f\left( g(x) \right) = 7^{g(x)} = 7^{\log_{5}x}.\]

\[\textbf{г)}\ g\left( g(x) \right) = \log_{5}\left( g(x) \right) =\]

\[= \log_{5}{(\log_{5}x}).\]

\[\textbf{д)}\ g\left( \varphi\left( f(x) \right) \right) = \log_{5}{(\varphi\left( f(x) \right)} =\]

\[= \log_{5}\left( \left( f(x) \right)^{2} \right) = \log_{5}\left( \left( 7^{x} \right)^{2} \right) =\]

\[= 2x\log_{5}7.\]

\[\textbf{е)}\ \varphi\left( g\left( f(x) \right) \right) = \left( g\left( f(x) \right) \right)^{2} =\]

\[= \left( \log_{5}\left( f(x) \right) \right)^{2} = \left( \log_{5}7^{x} \right)^{2} =\]

\[= \left( x\log_{5}7 \right)^{2}.\]

\[\textbf{ж)}\ f\left( g\left( \varphi(x) \right) \right) = 7^{g\left( \varphi(x) \right)} =\]

\[= 7^{\log_{5}\left( \varphi(x) \right)} = 7^{\log_{5}x^{2}}.\]

\[\textbf{з)}\ f\left( g\left( f(x) \right) \right) = 7^{g\left( f(x) \right)} =\]

\[= 7^{\log_{5}\left( f(x) \right)} = 7^{\log_{5}7^{x}} = 7^{x\log_{5}7}.\]