In class we learned the following variant of the Implicit Function Theorem:
Suppose $f:U \to \mathbb{R}^{n-k}$, where $U \subseteq \mathbb{R}^n$, is such that $Df(p)$ has full row rank for all $p \in U$. Then there exist diffeomorphisms $\alpha, \beta$ s.t. $\alpha: U' \to U$ and $\beta: F(U) \to W$, and $\beta \circ f \circ \alpha = \pi$, the normal orthogonal projection $\mathbb{R}^n \to \mathbb{R}^{n-k}$.
My Question: Why do we need $\beta$? WLOG suppose that the first $n-k$ columns of $Df(p)$ are linearly independent. Then the function $F:(x_1, \dotsc, x_n) \mapsto (f_1, \dotsc, f_{n-k}, x_{n-k+1}, \dotsc, x_n)$ has an inverse in a neighborhood of $p$, so
$$f \circ F^{-1}: (f_1, \dotsc, f_{n-k}, x_{n-k+1}, \dotsc, x_n) \mapsto (f_1, \dotsc, f_{n-k}).$$
So why do we need $\beta$?