Question

已知函数f(x)=把函数g(x)=f(x)-x的零点按从小到大的顺序排列成一个数列,则该数列的通项公式为(　　)

A.a_n=         B.a_n=n-1

C.a_n=n(n-1)         D.a_n=2ⁿ-2

Answer 1

B.当x≤0时,g(x)=2^x-1-x,令g(x)=0,得x=0.

当0<x≤1时,-1<x-1≤0,g(x)=f(x-1)+1-x=2^x-1-x,令g(x)=0,得x=1,

当1<x≤2时,0<x-1≤1,-1<x-2≤0,

g(x)=f(x-1)+1-x=f(x-2)+2-x=2^x-2+1-x,

令g(x)=0,得x=2.

依次类推,得到函数g(x)的零点从小到大排列为0,1,2,3,4,…,故选B.