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My book says that if functions $f$ and $g$ are both onto then $f\circ g$ and $g\circ f$ may or may not be onto.

Why is this so? Would someone please help me understand this, maybe with an example or diagrammatically? My book states that$ f\circ g$ and $g\circ f$ may or may not be onto. I think this might be related to the domains and codomains of g and f which may not be equal. I am aware that there is a similar looking question and therefore I'm clarifying mine. With respect to the domains and codomains would someone please explain why the composite functions may or may not be onto?

Hema
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    If you want to discuss surjectivity you have to specify the domains and ranges of both $f$ and $g$. Otherwise your question is meaningless. There is no "intuitive way" around this. – Christian Blatter Sep 11 '18 at 19:02
  • @ChristianBlatter my question is actually about how fog and gof may or may not be surjective for various domains and ranges of f and g. – Hema Sep 12 '18 at 11:37

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The composition of two surjective functions is always surjective: Let $f: X \to Y$ and $g: Y \to Z$ be functions and $z \in Z$. Then since g is surjective, there exists $y$ such that $g(y)=z$ and similarly there exists $x$ such that $f(x)=y$. Then $g(f(x))=z$ and your composition is surjective.

You can do the same argument for injective functions.

Gabriel Micolet
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  • The answer is good. The only 'catch' I can think of that could make the original premise correct would be if $f: X \to Y_1$, and $g: Y_2 \to Z$, with $Y_1,Y_2$ related but not identical. – Ingix Sep 11 '18 at 13:12
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    I have actually edited and expanded the question, I just wanted to let you know Sir. – Hema Sep 11 '18 at 13:20