# Math Help - Rank of Functions

1. ## Rank of Functions

Let $f:{M}\rightarrow{N}$ and $g:{N}\rightarrow{K}$ be two differentiable functions and $M$, $N$ and $K$ be manifolds. If the rank of $g$ at $f(P)$ is equal to the dimension of $N$, show that $f$ and $g\circ f$ have the same rank at $P$.

Something tells me that I have to use the fundemental equation of dimension, but I can't work it out.

2. The "rank" of f:A-> B is the dimension of f(A) as a subspace of B. Since the rank of g:B-> C is the same as the dimension of C, g(B) is all of C and, in particular, g(f(A)) has the same dimension as f(A).