Question

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

• A. $f\left(x\right)$ is odd
• B. $f\left(x\right)$ and $f^{-1}\left(x\right)$ may not be symmetric about the line $y=x$
• C. $f\left(x\right)$ may not be odd
• D. None of these

Answer 1

The correct option is A $f\left(x\right)$ is odd

Since $f\left(x\right)$ and $f^{-1}\left(x\right)$ are symmetric about the line $y=-x$, if $(\alpha ,\beta )$ lies on $y=f\left(x\right)$, then $(-\beta ,-\alpha )$ lies on $y=f^{-1}\left(x\right)$.
Therefore, $(-\alpha ,-\beta )$ lies on $y=f\left(x\right)$.
Hence, $f\left(x\right)$ is an odd function.