Defining an inverse function

Question

Consider the map $f:\mathbb{R}^n\to\mathbb{R}^{n-1}$ that maps $x\mapsto y$ defined as \begin{align} f(x_1,x_2,...,x_n)=(y_1,y_2,...,y_{n-1}) \end{align} where $y_i = x_i - x_n, \forall i \in \{1,...,n-1\}$. Obviously, this map is not invertible. Can we "define" an inverse for $f$? For instance, we can define \begin{align} f^{-1}(y_1,y_2,...,y_{n-1})=(x_1,x_2,...,x_{n-1},0). \end{align} where $x_i = y_i, \forall i \in \{1,...,n-1\}$. Can we say that this "defined" inverse function is smooth? If yes, can we conclude that the restriction of $f$ to $\mathbb{R}^n\times\{0\} $ is a homeomorphism?