Question

15．已知函數(shù)f（x）=$\left\{\begin{array}{l}{x+1，x≤0}\\{lo{g}_{2}x，x＞0}\end{array}\right.$ 則函數(shù)y=f[f（x）+1]的零點(diǎn)個(gè)數(shù)是4．

Answer 1

分析 求出f（x）的零點(diǎn)，設(shè)為x₀，令f（x）+1=x₀即可解出y=f[f（x）+1]的零點(diǎn)．

解答 解：令f（x）=0得$\left\{\begin{array}{l}{x+1=0}\\{x≤0}\end{array}\right.$或$\left\{\begin{array}{l}{lo{g}_{2}x=0}\\{x＞0}\end{array}\right.$，解的x=-1或x=1．
即f（x）有兩個(gè)零點(diǎn)x=1，x=-1．
令f[f（x）+1]=0，則f（x）+1=1或f（x）+1=-1．
即f（x）=0或f（x）=-2．
當(dāng)f（x）=0時(shí)，x=±1，
當(dāng)f（x）=-2時(shí)，$\left\{\begin{array}{l}{x+1=-2}\\{x≤0}\end{array}\right.$或$\left\{\begin{array}{l}{lo{g}_{2}x=-2}\\{x＞0}\end{array}\right.$，解的x=-3或x=$\frac{1}{4}$．
綜上，f[f（x）+1]共有4個(gè)零點(diǎn)．
故答案為：4．

點(diǎn)評(píng) 本題考查了函數(shù)的零點(diǎn)計(jì)算，屬于中檔題．