Step 1: Use monotonic growth.
For $x \geq 1$, the derivative $f'(x)=2x+1$ is positive. So $f$ is strictly increasing on the naturals. A strictly increasing function never repeats a value, so it is one-one.
Step 2: Look at the range.
The outputs are $3, 7, 13, 21, \ldots$. The smallest value is $f(1)=3$. The numbers 1 and 2 are in the codomain but are not in the range.
Step 3: Conclude.
The range is a proper subset of $N$, so $f$ is not onto. Together with one-one, this matches option 2.
Step 4: Cross check with small values.
Take $f(1)=3$, $f(2)=7$, $f(3)=13$, $f(4)=21$. The outputs are all different and they keep growing, which agrees with one-one. The values 1 and 2 never appear. So option 1 (onto) and options 3 and 4 (many-one) all fail.
Final Answer:
One-one but not onto.
\[ \boxed{\text{Option 2}} \]