Suppose you have functions $f$ and $g$ such that $f: Y \rightarrow T$ and $g: X \rightarrow Y$. I having a tough time understanding this?  I've learned about onto and 1 to 1 functions from a basic perspective, but I'm having a hart time understanding where to even start with this.  
(a) Suppose (f o g) is onto.  Claim g is onto. Prove or Disprove.
(b) Suppose (f o g) is onto.  Claim f is onto. Prove or Disprove.
(c) Suppose (f o g) is one to one.  Claim g is one to one.  Prove or Disprove 
(c) Suppose (f o g) is one to one.  Claim f is one to one.  Prove or Disprove 
We've been given a hint of one is true for onto, and one is true for one to one. 

If (f o g) is onto, that means that for any x value that enters into g, there is a resulting value of g AND that each of these g values has a corresponding f output. This means that f's outputs are the ones that must have be fulfilled by all of the given inputs. This means that between (a) and (b), the claim that f is onto can be proven, where g does not have to be onto. 
However, if (f o g) is one to one, that means that each x value will map to exactly one value of g, and these g values will in turn each map to exactly one value of f. If any inputs were to map to the same g value, then these inputs would also map to the same f value and the composition would not be one to one, thus g must be one to one and (c) must be true. However, f could have inputs that are not members of g and are not one to one; this would not prevent the composition (f o g) from being one to one - so (d) cannot be proven. 
This does not provide proofs, but I hope that it does help you make sense of how one to one and onto functions apply with composed functions. 
