Frankly, you seem to have very little interest in these problems- you are not even copying them correctly! (Or your text book is just really really bad.) You cannot talk about "nullity of f and g" when there are NO functions called "f" or "g" in the problem. Since you do talk about linear functions S and T, I would think that you meant "the composition of S and T" but that still makes no sense. S is in L(V, W) so S maps vectors in V to vectors in W. T is also in L(V, W) so, again, maps vectors in V to W. To be able to compose ST, we would have to have T in L(V, W) and S in L(W, Z) for some vector space Z. To be able to compose TS, we would have to have S in L(V, W) and T in L(W, Z).
If you meant "S is in L(V, W) and T is in L(W, Z), show that nullity(TS) is less than or equal to nullity(T)+ nullity(S)", then take v in nullity(S). What is S(v)? What is TS(v)?