Question

Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto. (Hint: Consider f(x) = x + 1 and g(x) = { x - 1 if x > 1
1 if x = 1 }

Answer 1

example -1 :- Let f(x) = x + 1 and $g(x)=\left\{\begin{array}{ll}x-1&if\:x\geq1\\1&if\:x=1\end{array}\right.$
Let f(x) = y = x + 1
so, x = y - 1
for y = 1 , x = -1 { it isn't a natural number }
hence, function f is not onto.
now when x > 1 , gof(x) = g(f(x)) = x + 1 - 1 = x
gof(x) = x , is identity function . hence it is an onto function.

example - 2 :- Let f : N → N be defined as f(x) = x + 2
and g : N → N be defined as $g(x)=\left\{\begin{array}{ll}x-2&if\:x>2\\2&if\:x=2\end{array}\right.$
here you can also observed f is not onto but gof is onto function.