Creating a map from $x \in [-1,1]$ to $y\in\mathbb{R}^2$ such that $y_1 - y_2$ is 1-1 w.r.t. $x$

Question

The subject line pretty much said what I need to do, that is I want to create a mapping from $x \in [-1,1]$ to $y \in \mathbb{R}^2$ with the following property: for each $x, x' \in [-1,1]$ $( y_1(x) - y_2(x) ) \ne ( y_1(x') - y_2(x') )$. Obviously I can do this parametrically, but I'm wondering if there's any nonparametric property that I can invoke that would imply my condition. It seems very unlikely to me, but you guys are much smarter than me, so I thought I'd give it a try...

Answer 1

Define your function as $$f(x)=(x,1)$$ Note that if $x\ne x'$, then $1-x \ne 1-x'$

Thus this function satisfies your condition.

Answer 2

Hint.- Consider $x\to (e^{x+1}, e^x)$ and the fact that $g(x)=e^{x+1}-e^x$ is one-to-one.