If $g\circ f$ is injective, what can you say about injectivity of $f$ and $g$?

Question

If $g\circ f$ is injective, what can you say about injectivity of $f$ and $g$? I tried but cannot able to get to the solution. Please help.

Answer 1

If $g\circ f$ is injective then $f$ should be injective as well.

Proof:

First we recall the definition of an injective function. We say that a function h ś injective if $$\forall a,b (\; h(a) = h(b) \; \Rightarrow a = b \; )$$

Suppose $g\circ f$ is injective.

We want to prove that f is injective.

Let a and b such that $$f(a) = f(b)$$

So, we have:

$$g\circ f(a) = g\circ f(b)$$.

As $g\circ f$ is injective, then we can conclude that:

$$ a=b $$.

So, $f$ is necessarily injective.

Answer 2

HINT: One of the function has to be injective, the other doesn't. Look at the case where $f$ is not surjective.