$\displaystyle y^3 + y = e^x -x + 9$

a) Show that for every x, there exists only exactly one y.

b) Let the solution $\displaystyle y$ in (a) be $\displaystyle y = f(x).$ Show that this function is $\displaystyle C^1$ locally.

c) With the previous two answers in mind, how does it now follow that $\displaystyle f \in C^1(R) $

d) What is the domain and range of f.