I think that $T_f(x)=f'(x)$ is invertible. This seems likely because it is a differential operator, and the inverse of a differential operator is the integral operator (though I'd like more explanation on this point).
If that is the case, would $T^{-1}$$(x^4+x^3+x^2+x+1) $ = $\frac{x^5}{5}$ + $\frac{x^4}{4}$ + $\frac{x^3}{3}$ + $\frac{x^2}{2} + x + c$?
In general no, and in fact you've exposed why already: if $f'(x)=g(x)$ then $(f+c)'(x)=g(x)$, for any constant function $c$.