If f is surjective and g is injective, what is $f\circ g$ and $g\circ f$? Say I have $f=x^2$ (surjective) and $g=e^x$ (injective), what would $f\circ g$ and $g\circ f$ be? (injective or surjective?)
Both $f$ and $g : \mathbb{R} \to \mathbb{R}$.
I've graphed these out using Maple but I don't know how to write the proof, please help me!

You should specify the domains and codomains of your functions.
I guess that 
$f:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$, but there are some other natural definitions you could make.
You can write down the compositions explicitly:
$f\circ g:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ has $x\mapsto (e^x)^2=e^{2x}$.
This is injective (since $x\mapsto e^x$ is injective) and not surjective, since 0 is not in the image.
$g\circ f:\mathbb{R}\rightarrow\mathbb{R}$ $x\mapsto e^{x^2}$ is neither injective, nor surjective.
