Find an example of functions $f:A\to B$ and $g:B\to C$ such that $f$ and $g\circ f$ are both injective, but $g$ is not injective.
So If I understand this correctly,
- Need a function $f$ that is injective and that will also make $g$ injective when plugged in during $g\circ f$.
- Need a function $g$ that is not injective on its own
- The range of $f$ must be a subset of the domain of $g$
I tried thinking along the lines of using variations of $f(x)=x$ and $g(x)=x^2$ but all those leave my composite function as non injective
I've also been using $x\in\Bbb R$ so as to keep the range of $f$ and domain of $g$ the same.
Any suggestions of where to go with this? Thank you
Edit: Thank you everyone the answers were very helpful in understanding the problem and concepts better