Let $f : L\rightarrow M$ be a homeomorphism onto its image (aka a topological embedding), and let $q : M \rightarrow N$ be a topological quotient map.
Suppose $(q \circ f) : L\rightarrow N$ is injective. Then is $q \circ f$ also a homeomorphism onto its image?
I'm not sure how to go about proving the affirmative, nor how to find a counterexample. Would anyone have any suggestions on how to proceed?
No, $q\circ f$ need not be a homeomorpism onto its image.
One thing to note about this question is that it's equivalent to asking if $q\colon M\to N$ is a quotient map and $A\subseteq M$ is such that the restriction $q|_A$ is injective, does it follow that $q|_A$ is a homeomorphism? (To see that its equivalent, note that its certainly a special case if we view $L\colon A\to M$ as the inclusion map, and conversely, the original statement regarding $L$ reduces to the new statement for $A=f(L)$.)
But in spite of the fact that you only need to check for inclusions of subsets of $M$, it's still false. For a simple counter-example, let $M=\mathbb R^2$, let $q\colon \mathbb R^2\to \mathbb R$ be projection onto the $x$-axis, and let $A=\{(x,\frac{1}{x})\mid x\in\mathbb R\backslash \{0\}\}\cup\{(0,0)\}$ (or, more generally, let $A$ be the graph of any discontinuous function defined on all of $\mathbb R$).