The Answer to the Question
is below this banner.
Here's the Solution to this Question
Since in question domain and codomain of both functions were not exactly specified, answer will be given in assumption of domain and codomain of both functions being subsets of .
a)Yes. Since codomain of g ◦ f function is same as codomain of function g due to definition of function composition. If and then .
For example if f: f(x)=|x| and g:g(x)=|x|. This will result in function and obviously for every element y in Z+ exist at least one element x in Z such as g ◦ f(x)=y
b)No. As said above function g defines codomain of composition, and if where is not empty subset such as .
c)Yes. if for example f: f(x)=2x and g: g(x)=|x|. g is not on-to-one function, since exists x1=1 and x2=-1 which make statement false. But function g ◦ f is function and now for every x1 and x2