Question

The mapping $f:\mathrm{\mathbb{N}}\to \mathrm{\mathbb{N}}$ is given by $f\left(n\right)=1+n^2$, $n$ belongs to $\mathrm{\mathbb{N}}$, where $\mathrm{\mathbb{N}}$ is the set of natural numbers, is :

• A. one-one and onto
• B. onto but not one-one
• C. one-one but not onto
• D. neither one-one nor onto

Answer 1

The correct option is C

one-one but not onto

Explanation for the correct option

Given function, $f\left(n\right)=1+n^2$

If $n=1$, value of $f\left(n\right)=2$

If $n=2$, value of $f\left(n\right)=5$

If $n=3$, value of $f\left(n\right)=10$

Thus,$f\left(n\right)$ is one-one.

Bur clearly, range $\ne$ co-domain,, range$\subset$co-domain

So, $f\left(n\right)$ is not onto.

Hence, $\mathrm{Option}\left(\mathrm{C}\right)$ is the correct answer.