composite functions and their injectivity and surjectivity

Here is the question:

let f : A B and g : B C be functions.

i) show that if g f is injective, then f is injective

ii) show that if g f is surjective, then g is surjective.

Here is what I have so far:

Now assume that is injective. then is one to one and

So from here I know the goal is to show that . But how?? my guess is that inverse functions are involved?? maybe not.

As far as part b goes, I'm thinking that it is a similar process, almost reversed. but I'm still somewhat confused on this part. thank you for the help!