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Suppose the function $g$ and $f$ are one-to-one. Is $f \circ g$ one-to-one?

Suppose $f \circ g$ is one-to-one, are the function $g$ and $f$ one-to-one?

Suppose $f \circ g$ is onto, are the function $g$ and $f$ onto?

Suppose the function $g$ and $f$ are onto. Is $f \circ g$ onto?

I was trying to think of examples to respond to those questions, but I couldn't think of anything.

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    See, for example, [Sufficient / necessary conditions for $f\circ g$ being injective, surjective or bijective](http://math.stackexchange.com/questions/208756/sufficient-necessary-conditions-for-f-circ-g-being-injective-surjective-or) and [If $g\circ f$ is the identity function, then which of $f$ and $g$ is onto and which is one-to-one?](http://math.stackexchange.com/questions/75880/if-g-circ-f-is-the-identity-function-then-which-of-f-and-g-is-onto-and-w/) (and the links to other questions or online-resources given there).2012-11-04

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