Bijective, Surjective and Injective Functions

$f$ is injective if $f(x) = f(y)$ implies that $x=y$. Equivalently, if $x \ne y$ then $f(x) \ne f(y)$. That is, a function is injective if it maps distinct points to distinct points, so that a point is determined by its image (i.e. there is a well-defined inverse on a subset of the codomain).

It's usually easier to work with the definition that $f$ is injective if $f(x) = f(y) \Rightarrow x=y$.

So if $f$ and $g$ are injective and $f(g(x)) = f(g(y))$, then...?

A function is surjective if every element of the codomain can be written as $f(x)$ for some $x$. For example, consider $f(x)=x^2$ on the real numbers. This isn't surjective, since you can't write -1 as $x^2$ for any real value of $x$. But the function $f(x)=x^3$ is surjective, since if $y$ is any real number and $x=\sqrt[3]{y}$ then $f(x)=y$, so we can write every real number in the form $f(x)$.

You might also want to have a look at http://gowers.wordpress.com/2011/10/11/injections-surjections-and-all-that/