Question:
Prove that the function $f: N \rightarrow N: f(n)=\left(n^{2}+n+1\right)$ is one - one but not onto.
Solution:
In the given range of $N f(x)$ is monotonically increasing.
$\therefore f(n)=n^{2}+n+1$ is one one.
But Range of $f(n)=[0.75, \infty) \neq N$ (codomain)
Hence, $f(n)$ is not onto.
Hence, proved that the function $f: N \rightarrow N: f(n)=\left(n^{2}+n+1\right)$ is one - one but not onto.
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