📝 题目
例 5 求函数 $y = {2x} + \left| {2 - x}\right| \left( {-\infty < x < + \infty }\right)$ 的反函数,并画出它的图形.
💡 答案与解析
解 $\forall y$ 视 $x$ 为未知数,解方程 ${2x} + \left| {2 - x}\right| = y$ . 为了去掉绝对值, 将方程改写为
$$ y = \left\{ {\begin{array}{ll} x + 2 & \left( {x \leq 2}\right) , \\ {3x} - 2 & \left( {x > 2}\right) \end{array} \Rightarrow x = \left\{ \begin{array}{ll} y - 2 & \left( {y \leq 4}\right) , \\ \frac{y + 2}{3} & \left( {y > 4}\right) \end{array}\right. }\right. $$
$$ \overset{x,y\text{ 互换 }}{ \Rightarrow }y = \left\{ \begin{array}{ll} x - 2 & \left( {x \leq 4}\right) , \\ \frac{x + 2}{3} & \left( {x > 4}\right) . \end{array}\right. $$
如图 1.2 所示.
\begin{center} \includegraphics[max width=0.3\textwidth]{images/002.jpg} \end{center} \hspace*{3em}
图 1.2