$g:[0,1]\rightarrow R $ is a strictly increasing continous function with $g(0)=0$ and $c(a)=g^-(g(1)-g(a))$ then $g(a)=?$

Question

Let $g:[0,1]\rightarrow R$ be a strictly increasing continous function with $g(0)=0$ and $c:[0,1]\rightarrow [0,1]$ be a continous function.

Let $c(a)=g^-(g(1)-g(a))$ for some $a\in [0,1]$ Then $g(a)=?$

I did try to show that $a$ is fixed point of $c$ but I couldn't. I strongly beleive that the question is lacking information as nothing much has been said about $c$ and $a$ .Still I want to confirm from the experienced people here.

Please share your views or ideas before closing the question (if required to do so)