If g(f(x) is injective I need to determine if f is injective. If it is false, I need to give a counterexample.
I was told that using arrow diagrams helps.
Can some one help me get started?
assume $\displaystyle f(x_1) = f(x_2)$. then $\displaystyle g(f(x_1)) = g(f(x_2))$. but that means $\displaystyle (g \circ f)(x_1) = (g \circ f)(x_2)$. since $\displaystyle g \circ f$ is injective, this means $\displaystyle x_1 = x_2$. thus, $\displaystyle f$ is injective