Question

if $f:R\to R$ is an invertible function such that $f\left(x\right)$ and $f^{-1}\left(x\right)$ 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

Answer: (A)

$f\left(x\right)$ is odd

From the given condition we know that $f\left(x\right)$ is invertible.
Since it is symmetric about $y=-x$,
$f(-x)=-f\left(x\right)$
Thus, $f\left(x\right)$ is odd.
Hence, option 'A' is correct.