If $f:R\to R$ is an invertible function such that f(x) and $f^{-1}\left(x\right)$ are symmetric about the line $y=-x,$ then
 

Since $f\left(x\right)and{f}^{-1}\left(x\right)$ are symmetric about the line y = -x

If $\left(\alpha ,\beta \right)$ lies on y = f(x) then $\left(-\alpha ,-\beta \right)$ lie on $y=f^{-1}\left(x\right)$

$\Rightarrow \left(-\alpha ,-\beta \right)$ lies on $y=f\left(x\right)\Rightarrow y=f\left(x\right)$ is odd.