Query about inverse function.

Question

"I have read inverse of f(x) is symmetrical about Y=X ".

But what about inverse of Y = -X$^{3}$.It's answer will be -(X) $^{1/3}$. Which is symmetrical about Y =-X.

But nowhere it's mentioned that it can be also reflected about Y = -X.

Where am i wrong?

Answer 1

The point here is that the function is an odd function, which means that any point on that function is reflected in $(0,0)$. Now, when you reflect a point in $(0,0)$ AND in $y=x$ you have effectively reflected that point in $y=-x$.

Answer 2

You are not wrong. If $f(x)=-x^3$, then the graph of $f$ is symmetric about $y=x$ and(!) about $y=-x.$

Answer 3

That particular function has the additional property of being odd ($f(-x)=-f(x)$), which creates more symmetry. This also works with $f(x)=x^3$, though you will find it less visually striking.