11.设 (x)=dfrac (x)(sqrt {1+{x)^2}} 求 [ f(x)] , f[ f(x)] .

步骤 1：计算 $f[f(x)]$
首先，我们已知 $f(x)=\dfrac {x}{\sqrt {1+{x}^{2}}}$。为了计算 $f[f(x)]$，我们需要将 $f(x)$ 代入 $f$ 的定义中。即，将 $f(x)$ 的表达式代入 $x$ 的位置。
$$f[f(x)] = f\left(\dfrac {x}{\sqrt {1+{x}^{2}}}\right) = \dfrac {\dfrac {x}{\sqrt {1+{x}^{2}}}}{\sqrt {1+\left(\dfrac {x}{\sqrt {1+{x}^{2}}}\right)^2}}$$
步骤 2：化简 $f[f(x)]$
化简上述表达式，我们得到：
$$f[f(x)] = \dfrac {\dfrac {x}{\sqrt {1+{x}^{2}}}}{\sqrt {1+\dfrac {x^2}{1+{x}^{2}}}} = \dfrac {\dfrac {x}{\sqrt {1+{x}^{2}}}}{\sqrt {\dfrac {1+{x}^{2}+x^2}{1+{x}^{2}}}} = \dfrac {\dfrac {x}{\sqrt {1+{x}^{2}}}}{\sqrt {\dfrac {1+2{x}^{2}}{1+{x}^{2}}}} = \dfrac {x}{\sqrt {1+2{x}^{2}}}$$
步骤 3：计算 $f\{ f[ f(x)] \}$
接下来，我们计算 $f\{ f[ f(x)] \}$，即 $f\left(\dfrac {x}{\sqrt {1+2{x}^{2}}}\right)$。
$$f\{ f[ f(x)] \} = f\left(\dfrac {x}{\sqrt {1+2{x}^{2}}}\right) = \dfrac {\dfrac {x}{\sqrt {1+2{x}^{2}}}}{\sqrt {1+\left(\dfrac {x}{\sqrt {1+2{x}^{2}}}\right)^2}}$$
步骤 4：化简 $f\{ f[ f(x)] \}$
化简上述表达式，我们得到：
$$f\{ f[ f(x)] \} = \dfrac {\dfrac {x}{\sqrt {1+2{x}^{2}}}}{\sqrt {1+\dfrac {x^2}{1+2{x}^{2}}}} = \dfrac {\dfrac {x}{\sqrt {1+2{x}^{2}}}}{\sqrt {\dfrac {1+2{x}^{2}+x^2}{1+2{x}^{2}}}} = \dfrac {\dfrac {x}{\sqrt {1+2{x}^{2}}}}{\sqrt {\dfrac {1+3{x}^{2}}{1+2{x}^{2}}}} = \dfrac {x}{\sqrt {1+3{x}^{2}}}$$