Isometry Help!

Question

I need some help with these isometry problems. 

 

 

Is there an isometry $f$ of the line so that $f(-3)=-2$ and $f(1)=2$?


{ 2.} $f$ is an isometry of the line so that $f(-3)=4$ and $f(2)=9$. Find $f(0.5)$

Answer 1

1. Yes, you need to check whether transformation preserves distances. Meaning, -3 to 1 --> distance is 4, and -2 to 2 --> distance is again 4.
2. Since  $f$ is an isometry and $f(2)=9$ (you can pick either), we can conclude that the line shifts by 7 units, $f(x)=x+7$. Therefore, $f(0.5)=7.5$