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Let $X$ be a set, and let $f:X \longrightarrow X$ and $g:X\longrightarrow X$. Can you get a composition of $f$ and $g$ ($f\circ g$) that is onto (surjective), if $g$ is not onto?

I can't seem to find an example of such a function.

Thank you for your help.

Bernard
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royvicproust
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2 Answers2

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Let $g\colon \mathbb N\to\mathbb N$ be the successor function $g(n)=n+1$ and let $f\colon \mathbb N\to\mathbb N$ be the function $f(n)=n-1$ if $n>1$ and $f(1)=1$. Then $f\circ g$ is the identity (and hence is surjective) whereas $g$ is not (since $1$ is not in its range).

pre-kidney
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If $X$ is finite then no (since it is equivalent to being 1-1). Otherwise consider $X=\mathbb{N}$ and the functions $g(x)=2x$ and $f(x)=\lceil{x/2}\rceil$

Μάρκος Καραμέρης
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