Question

已知函数f(x)=2sin(2x-π/4),若f(x+a)为奇函数,a∈[0,2π),求a,若为偶函数,求a

Answer 1

函数f(x)=2sin(2x-π/4),若f(x+a)为奇函数,

y=sinx为奇函数

f(x+a)=2sin(2(x+a)-π/4)=2sin(2x+2a-π/4)

2a-π/4=kπ

2a-π/4=0 a=π/8

或2a-π/4=π a=5π/8

2a-π/4=2π a=9π/8

2a-π/4=3π a=13π/8

若为偶函数,

y=cosx为偶函数

2a-π/4=kπ+π/2

a=kπ/2+3π/4

k=-1 a=π/4

k=0 a=5π/4

k=2 a=11π/8