Question

1．設(shè)x∈R，定義符號(hào)函數(shù)sgnx=$\left\{\begin{array}{l}{1，x＞0}\\{0，x=0}\\{-1，x＜0}\end{array}\right.$，則（　　）

• A． |x|=x|sgnx|
• B． |x|=xsgn|x|
• C． |x|=|x|sgnx
• D． |x|=xsgnx

Answer 1

分析 去掉絕對(duì)值符號(hào)，逐個(gè)比較即可．

解答 解：對(duì)于選項(xiàng)A，右邊=x|sgnx|=$\left\{\begin{array}{l}{x，}&{x≠0}\\{0，}&{x=0}\end{array}\right.$，而左邊=|x|=$\left\{\begin{array}{l}{x，}&{x≥0}\\{-x，}&{x＜0}\end{array}\right.$，顯然不正確；
對(duì)于選項(xiàng)B，右邊=xsgn|x|=$\left\{\begin{array}{l}{x，}&{x≠0}\\{0，}&{x=0}\end{array}\right.$，而左邊=|x|=$\left\{\begin{array}{l}{x，}&{x≥0}\\{-x，}&{x＜0}\end{array}\right.$，顯然不正確；
對(duì)于選項(xiàng)C，右邊=|x|sgnx=$\left\{\begin{array}{l}{x，}&{x≠0}\\{0，}&{x=0}\end{array}\right.$，而左邊=|x|=$\left\{\begin{array}{l}{x，}&{x≥0}\\{-x，}&{x＜0}\end{array}\right.$，顯然不正確；
對(duì)于選項(xiàng)D，右邊=xsgnx=$\left\{\begin{array}{l}{x，}&{x＞0}\\{0，}&{x=0}\\{-x，}&{x＜0}\end{array}\right.$，而左邊=|x|=$\left\{\begin{array}{l}{x，}&{x≥0}\\{-x，}&{x＜0}\end{array}\right.$，顯然正確；
故選：D．

點(diǎn)評(píng) 本題考查函數(shù)表達(dá)式的比較，正確去絕對(duì)值符號(hào)是解決本題的關(guān)鍵，注意解題方法的積累，屬于中檔題．