# Give examples of two functions $$f: N \to N$$ and $$g: N \to N$$ such that $$g\;o\;f$$ is onto but $$f$$ is not onto.

Toolbox:
• A function $f:A \to B$ is onto if for $y \to B$ then exists unique $x \in A$ such that $f(x)=y$
• We define two function $f:N \to N$ and $g: N \to N$ such that gof is onto $gof(x)=gof(y)$
Given $g(x)= \left\{ \begin{array}{1 1} x-1 & \quad x > 1\\ 0 & \quad x=1 \end{array} \right.$
Consider $x=1 \in$ Codomain of $g$. It is clear that there exists 2 element $x=1, x=2$ such that $g(x)=1$
Therefore $f$ is not onto
Consider $f(x)=x+1$
$\Rightarrow gof(x)=g(f(x))$ $=g(x+1)$ $=x+1-1 = x$
Since $x \in N;x+1 > 1$
$\Rightarrow gof$ is onto. since for $y \in N$ then exists $x \in N$ such that $gof(x)=y$
edited Mar 20, 2013