If $f^3(x)+f(x)=x+e^x$ then $f(x)=x$ has at least one solution in the interval $(1,3)$. Let an $f: \mathbb{R} \rightarrow \mathbb{R}$ such as $f^3(x)+f(x)=x+e^x$.  Show that the equation $f(x)=x$ has at least one solution in the interval $(1,3)$. 
I was able to prove that $f$ is increasing and continuous but couldn't proceed more.
Thank you

$f^3(1)+f(1)=1+e^1>3\Rightarrow f(1)>1$ similarly $f^3(3)+f(3)=1+e^3<30\Rightarrow f(3)<3$  thus $f(x)-x$ has at least a root in $(1,3)$.
