Is this function composed of non-invertible functions invertible?

Question

Consider a nonlinear function $f:\mathbb{R}^3 \rightarrow \mathbb{R}^2$. Next define $g:\mathbb{R}^4 \rightarrow \mathbb{R}^4$ as per: \begin{align} (y_1, y_2, y_3, y_4) = g(x_1, x_2, x_3, x_4) = (f(x_1,x_2,x_3), f(x_2, x_3, x_4)). \end{align} Can $g$ be invertible even though $f$ is not? I don't see why not, even though it at the same time sounds counter-intuitive to me. Edit: $f$ shouldn't "drop" any of the inputs.

Answer 1

Let $f$ drop the middle element. That is : $f(x, y, z) = (x, z)$. Then, $g(x, y, z, w) = (f(x, y, z) , f(y, z, w)) = (x, z, y, w)$, which is invertible.