Inverse of a function intersecting at y =X line 
The inverse of a function intersects the function on $y=x$ line.

This is what I was taught. It works fine for $y=x^2, x^3$ , 
Eg $y = x^2$ meet $x= y^2$ at$ (1,1)$ but..
For a function like $ y =-x^3$
It seems to intersect at $ x+y = 0 , $
Why, is the first statement wrong.
Also can it so happen , that an inverse of a function meets the function on a point other than on line $ y=±x $??

Consider the curve $y=1-x$. It's inverse is $y=1-x$, i.e. it is self inverse. This means it intersects all along its curve, despite only intersecting $y=x$ once.
Now suppose a curve $y=f(x)$ intersects the line $y=x$ at $x_0$. This means that $$y_0=f(x_0)=x_0.$$
Applying $f$ to both sides yields
$$
f(y_0)=f(x_0)=x_0,
$$
and hence the inverse of the curve intersects the curve at its intersection with $y=x$.
