Injective and Bijective functions [closed]

1. Suppose that $$f: A→B$$ and $$g:B→C$$ are functions such that $$g◦f$$ is injective. Prove that $$f$$ must be injective.

2. Construct a bijective function $$f:R→ (R\setminus \{0\})$$. Prove that your function is actually a bijective function.

Can someone help me on how do I prove it?

closed as off-topic by Scientifica, Gibbs, José Carlos Santos, Saad, Xander HendersonOct 4 '18 at 11:18

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• For two, try the function $$f(x)=\begin{cases}x&x\notin\mathbb N\cup\{0\}\\x+1&x\in\mathbb{N}\cup\{0\}\end{cases}$$ – Don Thousand Oct 3 '18 at 22:00

Given a value $$f(x) = f(y) \in B$$, since $$g$$ is a function,

$$g(f(x)) = g(f(y).$$

By the definition of injection of $$g\circ f$$, $$x=y$$.

So we have proven that $$f(x)=f(y) \implies x = y,$$

i.e. $$f$$ is injective.

Suppose f is not invective.

There exists some $$x,y$$ in $$A$$ such that $$f(x)= f(y)$$

It which case $$(g\circ f)(x) = (g\circ f)(y)$$