函数f(x)=arcsin(sinx)可以用分段函数表示吗

f(x)=arcsin(sinx)
当x∈[-π/2,π/2]时,
arcsin(sinx)=x
当x∈(π/2,3π/2]时,
π-x∈[-π/2,π/2)
sinx=sin(π-x)
∴arcsin(sinx)=π-x
当x∈[2kπ-π/2,2kπ+π/2],k∈Z
x-2kπ∈[-π/2,π/2],k∈Z
∴arcsin(sinx)=x-2kπ,
当x∈(2kπ+π/2,2kπ+3π/2],k∈Z
(2k+1)π-x∈[-π/2,π/2)
arcsin(sinx)=(2k+1)π-x
即
f(x)={ x-2kπ, x∈[2kπ-π/2,2kπ+π/2],k∈Z
{(2k+1)π-x, x∈(2kπ+π/2,2kπ+3π/2],k∈Z