Are the following immersions and embeddings? Are the following immersions, injective, embeddings?
(1) $f:\mathbb{R} \to \mathbb{R^2}$ where $f(t)=(t^2,t^3)$
(2) $f:\mathbb{R} \to S^1$ where $f(t)=(\cos t, \sin t)$
(3) $f:\mathbb{R} \to \mathbb{R^2}$ where $f(t)=(\cosh t,\sinh t)$
I thinking (1) and (3) are injective but (2) isn't since $(a^2,a^3)=(b^2,b^3) \Rightarrow a=b$ so (1) is injective. Similarly for (3).
How do I know are they an immersion or embedding? Do I first compute $Df$, if $Df$ is injective then $f$ is an immersion?
 A: Let's review some definitions:

A smooth function $f: M \to N$ is an immersion if $Df_p : T_pM \to T_{f(p)}N$ is an injective map for every $p \in M$.
A continuous function $f: M \to N$ is a topological embedding if it is a homeomorphism onto its image; i.e. if $f : M \to f(M)$ is a homeomorphism.
A smooth function $f: M \to N$ is an embedding if it is an injective immersion and a topological embedding.

So as you can see, for each function, you should check whether they're immersions and/or injections first, and if they're injective immersions, then you have to determine whether they're also a topological embedding.
You already determined that (1) is injective. However if you compute its derivative, you'll find that $Df_0$ is not injective. Thus it is not an immersion and cannot be an embedding.
You also figured out that (2) is not injective. Its derivative is $Df_t = (-\sin t, \cos t)$, which is an injective linear map for every $t$. Thus, it's an immersion, but since it's not injective, it can't be an embedding.
You said that (3) is injective, and if you compute its derivative, you'll find that it is also an immersion. The question remains then, is it an embedding? A quick sketch suggests that it is; I'll leave it up to you to determine why, with the following hint: Projecting onto the $y$-axis and then applying $\sinh^{-1}(t)$ is a left inverse for the map $f$.
