For $f:\mathbb R^{<0}\to\mathbb R$, $f(x)=2x^2-3$, find the values of a for which $f(a)=f^{-1}(a)$ Okay, i've got the answer for this with some luck I guess, however i'm still left wondering  specifically what this part of the question means:
"find the values of a for which $f(a)=f^{-1}(a)$"
My understanding of this is, that the question is asking me to find a value of a where the output and input given by the function $f$ are equal?
Could someone do a better job of explaining this to me please, thank you.

The values for which $f(x)=f^{-1}(x)$ must lie on the line $y=x$, since by the sheer definition of inverse function this is an axis of symmetry. This means that if a function and its inverse intersect, the points of intersection must lie on that line. Hence, you have to solve for which values of $x$ we have $f(x)=x$. So $2x^2-3=x$ and this leads to $x=-1$ or $x=1\frac{1}{2}$. Since your domain is the negative reals, you are left with $x=-1$.
