Give an example of functions f:A---->B and g:B----->C such that?

g o f is onto C, but f is not onto B.





g is one-to-one, but g o f is not one-to-one.





(f o g is a function compostion, f composed with g)

Give an example of functions f:A----%26gt;B and g:B-----%26gt;C such that?
For the first one, say that A={0,1}, B={0,1}, and C={0}


Then if f(x)=0 and g(x)=0,


f(x) is not onto B (no x is such that f(x)=1 and 1 is in B)


but g o f is onto C (range of g o f = {0} = C)





For the second one, you can do it real easily: If C=B and |B| %26gt; |A| and g(x)=x, then certainly g is one-to-one, but (g o f) is not one-to-one since its range is larger than its domain.