Question

Let a function ƒ : N →N be defined by
f(n) = \left\\{ \begin{array}{ll} 2n & n = 2,4,6,8,\ldots \\\ n - 1 & n = 3,7,11,15,\ldots \\\ \frac{n+1}{2} & n = 1,5,9,13 \end{array} \right.
then, ƒ is

• A. One-one but not onto
• B. Onto but not one-one
• C. Neither one-one nor onto
• D. One-one and onto

Answer 1

Answer: (D)

One-one and onto

Answer 2

The correct answer is (D) : One-one and onto
For n=1,5,9,13 n=1,5,9,13, $\frac{n+1}{2}$ yields all odd numbers.
When n=3,7,11,15,…n=3,7,11,15,…, n-1 is even but not divisible by 4.
For n=2,4,6,8,… n=2,4,6,8,…, 2n gives all multiples of 4.So, the range will be the set of all natural numbers.
Additionally, each value of n corresponds to a unique y, implying the function is one-to-one and onto.