Differential inverse, proof

Prove that the differentiable function $F(x,y,z)=\left ( f(x,y,z) ,g(x,y,z) , f(x,y,z)+g(x,y,z) \right)$ can never have a differentiable inverse.
My attempt : I notice that $\dim dom F=3$ and that $\dim Im F =2$. Hence $F$ is not injective, hence $F$ doesn't have an inverse, so it cannot have a differentiable inverse.
I'm not sure my attempt is good.
Maybe I should have done it using the Inverse Function Theorem, but if the Jacobian of $F$ is not invertible, I can't say anything whether the existence of an inverse of $F$, I believe. Hmm but according to my memory of Linear Algebra, if the matrix of a function is not invertible then the function is not injective so it doesn't have an inverse.
What do you think?