f(x)=sinx-｜cosx｜x属于R求其f(x)值域

Question

f(x)=sinx-｜cosx｜ x属于R 求其f(x)值域

Answer 1

sinx 和cosx 的最小正周期都是2π
所以 f(x) 有一个周期是2π
因此只需要求 [-π/2,3π/2]的值域（任选一个长度为2π的区间即可）
当 -π/2 ≤ x ≤ π/2 时,cosx ≥ 0
f(x) = sinx - cosx = √2 sin(x - π/4)
- 3π/4 ≤ x - π/4 ≤ π/4
- 1 ≤ sin(x - π/4) ≤ √2 /2
- √2 ≤ f(x) ≤ 1
当 π/2 < x ≤ 3π/2 时,cosx < 0
f(x) = sinx + cosx = √2 sin(x +π/4)
3π/4 < x + π/4 ≤ 7π/4
- 1 ≤ sin(x+π/4) < √2 /2
- √2 ≤ f(x) < 1
综上,f(x) 的值域为 [ - √2,1 ]