Let $f:[0,1]\rightarrow[0,1]$. $f$ is strictly increasing in $x$ if $x\in[0,x^*]$ and strictly decreases otherwise.
Suppose that I'm interested in finding the area of the domain where $f(x)\leq t$. That is, if $$t=f(x)=f(x+y)$$ for some $t$, $x\leq x^*$ and $x+y\geq x^*$, I want to find $y$.
Is there any closed form representation of $y$ in terms of $f$?
Further, is there any way that I can find $\frac{dy}{dt}$? Any suggestions?
Let's write $$f(x)=\begin{cases}f_1(x),x\in[0,x^*]\\f_2(x),x\in(x^*,1]\end{cases}$$ We know that both these branches are invertible (one is strictly increasing, one of them is strictly decreasing), so we can calculate $f_{1,2}^{-1}(y)$. If you sketch some graph of $f$, draw the line parallel to $x$ axis through $t$. You notice that it intersects $f_1$ at $(f_1^{-1}(t),t)$ and $f_2$ at $(f_2^{-1}(t),t)$. From these $$y=f_2^{-1}(t)-f_1^{-1}(t)$$ You can now take the derivative to get $\frac{dy}{dt}$.
In the same sketch of $f$, you can write the area under the curve in terms of integral along $y$ axis as well: $$A=\int_0^t(f_2^{-1}(y)-f_1^{-1}(y))dy-yt$$