Functional Relationships

Question

I have a function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the functional relationship $$f(x)+e^{f(x)}=2-x,\forall{x}\in{\mathbb{R}}$$ and I want to prove that $f$ is a monotonically decreasing function. Give me a hint!

Answer 1

Suppose that $f(y)\geqslant f(x)$. then $e^{f(y)}\geqslant e^{f(x)}$ and therefore $f(y)+e^{f(y)}\geqslant f(x)+e^{f(x)}$. But this means that $2-y\geqslant2-x$ and therefore that $y\leqslant x$. Can you take it from here?