Prove that the map $f:\mathbb{R}\to \mathbb{R}^2$ given $f(t)=1/2(e^t+e^{-t},e^t-e^{-t})$ is an embedding

Question

I'm not able to prove that the below map is embedding.

Prove that the map $f:\mathbb{R} \to \mathbb{R}^2$ given $f(t)=1/2(e^t+e^{-t},e^t-e^{-t})$ is an embedding and $M=f(R)$ is a $1$-manifold and graph it.

Answer 1

Hint:

You have( see hyperbolic functions) $$ f(t)=\left(\frac{e^t+e^{-t}}{2},\frac{e^t-e^{-t}}{2} \right)= (\cosh t,\sinh t) $$

and, since $\cosh^2 t-\sinh^2 t=1$ this is the parametric equation of an hyperbola.