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Question
Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.
Solution
Let us consider a function f : N → N given by f(x) = x +1 , which is not onto.
[This not onto because if we take 0 in N (co-domain), then,
0=x+1
⇒">⇒ x = - ∉ N ]
Let us consider g : N → N given by
g (x) = `{(x-1, ifx>1),(1,if x = 1):}`
Now, let us find (gof) (x)
Case 1 : x >1
(gof) (x) = g (f (x)) = g (x+1) = x+1−1 = x
Case 2 : x = 1
(gof) (x) = g (f (x)) = g (x+1) = 1
From case-1 and case-2, (gof) (x) = x, ∀x ∈ N,
which is an identity function and, hence, it is onto.
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