2. Indicate if the following statements are possible. Justify your answer. If the answer is “yes”, give a specific example of the functions. Let f: and g: Z be two functions. Use a diagram if that helps explain the function. a) Is it possible that f is not onto and g ◦ f is onto? b) Is it possible that g is not onto and g ◦ f is onto? c) Is it possible that g is not one-to-one and g ◦ f is one-to-one?
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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