已知函数f(x)=log1/2 |sin(x-派/4)|

f(x)的定义域为x≠2kπ+π/40< |sin(x-派/4)| ≤1又f(x)=log1/2 |sin(x-派/4)| 为单调递减函数,所以 |sin(x-派/4)| =1时取最小值,f(1）=0即f(x)的值域为[0,正无穷) f(-x)=log1/2 |sin(-x+π/4)| =log1/2 |sin(x-π/4)| =f(x),所以f(x)为偶函数 因为sin(x-派/4)的周期为2π,|sin(x-派/4)| ≥0,|sin(x-派/4)| 为sin(x-派/4)的半个周期所以|f(x)的周期为T=π f(x)=log1/2 |sin(x-π/4)| 为单调递减函数,当|sin(x-π/4)| 单减时,f(x)单增此时2kπ-π/2≤x-π/4≤2kπ,2kπ-π/4≤x≤2kπ+π/4当|sin(x-π/4)| 单增时,f(x)单减此时2kπ≤x-π/4≤2kπ+π/2,2kπ+π/4≤x≤2kπ+3π/4