Fun - Da - Mental : Graphs(2)

y=sin^-1(2x/1+x²)
Put x=tanθ
y=sin^-1(2tanθ/1+tan²Î¸)
y=sin^-1(sin2θ)
y=2θ
y=2tan^-1x

okay.
for the substitution, x=tan∂
we have ∂=tan^-1x
so as x belongs to -∞ to +∞, we will have ∂ from (-pi/2,pi/2)

now we know f(x)=sin^-1(sin2∂)
we have

f(x)=sin^-1(sin2∂)= 2∂ if -pi/2<2∂<pi/2 or -pi/4<∂<pi/4

= pi-2∂ if pi/2<2∂<pi or pi/4<∂<pi/2

= -2∂-pi if -pi<2∂<-pi/2 or -pi/2<∂<-pi/4

now just convert all ∂ to x substituting ∂=tan^-1x
to get f(x) in terms of x

cheers!!