How to prove that every rigid motion $F:\,\mathbb{R}\to \mathbb{R}$ is 1-to-1?

Question

A function $F:\,\mathbb{R}\to \mathbb{R}$ is a rigid motion if for all $x,y\in\mathbb{R}$ with $x\neq y$, $\vert x-y\vert = \vert F(x)-F(y)\vert$.

Using this definition of rigid motion, prove that every rigid motion is 1-to-1.

Any hint is highly appreciated.

Answer 1

Hint:

To show that $F$ is 1-to-1, you need to show that if $F(x)=F(y)$ then $x=y$.