Question

14．函数y=$\left\{\begin{array}{l}{2x，x≥0}\\{-{x}^{2}，x＜0}\\{\;}\end{array}\right.$的反函数是（　　）

• A． y=$\left\{\begin{array}{l}{\frac{x}{2}，x≥0}\\{\sqrt{-x}，x＜0}\\{\;}\end{array}\right.$
• B． y=$\left\{\begin{array}{l}{\frac{x}{2}，x≥0}\\{-\sqrt{-x}，x＜0}\\{\;}\end{array}\right.$
• C． y=$\left\{\begin{array}{l}{2x，x≥0}\\{\sqrt{-x}，x＜0}\end{array}\right.$
• D． y=$\left\{\begin{array}{l}{2x，x≥0}\\{-\sqrt{-x}，x＜0}\\{\;}\end{array}\right.$

Answer 1

分析 利用反函数的求法、分段函数的性质即可得出．

解答 解：∵y=$\left\{\begin{array}{l}{2x，x≥0}\\{-{x}^{2}，x＜0}\\{\;}\end{array}\right.$，x≥0时，由y=2x，解得x=$\frac{1}{2}y$，把x与y互换可得：y=$\frac{1}{2}$x；
x＜0，由y=-x²，解得x=-$\sqrt{-y}$，把x与y互换可得：y=$-\sqrt{-x}$．
∴函数y=$\left\{\begin{array}{l}{2x，x≥0}\\{-{x}^{2}，x＜0}\\{\;}\end{array}\right.$的反函数是y=$\left\{\begin{array}{l}{\frac{x}{2}，x≥0}\\{-\sqrt{-x}，x＜0}\end{array}\right.$．
故选：B．

点评 本题考查了反函数的求法、分段函数的性质，考查了推理能力与计算能力，属于中档题．