If $g\circ f$ is surjective, then $f$ is surjective.
46748
If $g \circ f$ is surjective, show that $f$ does not have to be surjective?
144589
Proving that if $f(x) \le g(x)$ then $f(x_0) \le g(x_0)$
56413
$f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
14994
Showing a function $f$ cannot be surjective
53551
$f(A\cap B)=f(A)\cap f(B)$ $\iff$ $f$ is injective.
58642
If $g \circ f$ is the identity function, then which of $f$ and $g$ is onto and which is one-to-one?
85768
The function $f:G \rightarrow G$ defined by $f(x)=x^2$ is a homomorphism iff G is abelian.
56278
$I$ and $J$ are coprime ideals iff $x \to (x + I, x + J)$ is surjective.
74657
Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$?
146152
Characterising functions $f$ that can be written as $f = g \circ g$?
32127
Example where $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective
126065
Example $g\circ f=id_A$ but $f\circ g\neq id_B$
145077
A continuous bijection $f:\mathbb{R}\to \mathbb{R}$ is an homeomorphism?
121993
Prove that $f$ is one-to-one iff $f \circ h = f \circ k$ implies $h = k$.
47709
If $n\mid m$ prove that the canonical surjection $\pi: \mathbb Z_m \rightarrow \mathbb Z_n$ is also surjective on units
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
14277
Proving $f(x)=x\cdot |x|$ is a bijection