If two maps are homotopic, are the images homotopy equivalent?

Question

My question is;

If two continuous maps $f,g:X\rightarrow Y$ are homotopic, are the images $Im(f),Im(g)$ homotopy equivalent?

Clearly, the converse is false.

If it is false, is there any condition to be true? (Such as $f$ or $g$ to be injective)

Answer 1

$X=Y=D^2$(closed 2-disc). Since $D^2$ is contractible, any map $f:D^2\to D^2$ is homotopic to constant map. Now consider $g_1:D^2\to D^2$ be a constant map and consider a surjective map $g:D^2\to S^1$ and define $g_2= i\circ g:D^2\to D^2$ where $i$ define the inclusion map of $S^1$ into $D^2$. Then image of $g_1$ is a point and image of $g_2$ is circle which are no thomotopically equivalnet.