千万别绕进去：自己复合自己的复合函数

一、题目

已知，$f(x)=\left\{\begin{array}{ll}2, & x>0, \\ \frac{1}{2}, & x=0, \\ -\frac{1}{2}, & x<0,\end{array}\right.$ 则 $f[f(x)]=?$

难度评级：

二、解析

$$
f(x)=\left\{\begin{array}{ll}2, & x>0 \\ \frac{1}{2}, & x=0 \\ -\frac{1}{2}, & x<0\end{array}\right. \Rightarrow
$$

$$
\left\{\begin{array}{ll}f(x)>0, & x \geqslant 0 \\ f(x)<0, & x<0\end{array} \Rightarrow\right.
$$

$$
f[f(x)]=\left\{\begin{array}{cc}2, & f(x)>0 \\ -\frac{1}{2}, & f(x)<0\end{array}\right. \Rightarrow
$$

$$
f[f(x)]=\left\{\begin{array}{ll}2, & x \geqslant 0 \\ -\frac{1}{2}, & x<0\end{array}\right.
$$

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