Question

Which of the following given functions is not a one-one function? (where 'n' belongs to natural numbers 'N').

• A. $f(x) = (\sqrt{n})(\sqrt{n})$
• B. $f(x) = n-1$
• C. $f(x) = 2^n$
• D. $f(x) = n^n$
• E. $f(x) = n-3$, if 'n' is odd $f(x) = 0$, if 'n' is even

Answer 1

Answer: (E)

Option (e)

Answer 2

For options (a)–(d), distinct $n$ yield distinct $f(n)$, but in option (e) even numbers map to 0, making it non-injective.

Option (a): $f(n)= (\sqrt{n})(\sqrt{n})=n$
This function is one-one since if $n_1 \neq n_2$ then $f(n_1) \neq f(n_2)$.

Option (b): $f(n)= n-1$
A linear function with slope 1 is one-one.

Option (c): $f(n)= 2^n$
The exponential function $2^n$ is strictly increasing and hence one-one.

Option (d): $f(n)= n^n$
For natural numbers, this function is strictly increasing, therefore one-one.

Option (e):

$$f(n)= \begin{cases} n-3 & \text{if } n \text{ is odd} \\ 0 & \text{if } n \text{ is even} \end{cases}$$

Here, every even $n$ gives $f(n)=0$. Thus, different even values (like $n=2, 4, 6, \dots$) have the same output. This violates the one-one property.