let f: N→N :f(x)={1/2(n+1), when n is odd n/2, when n is even. Then, f is

Question

Mark (√) against the correct answer in the following:

Let \(f:N→N:f(x)=\begin{cases}\frac{1}{2}(n+1), \text{ when n is odd}\\ \frac{n}{2},\text{when n is even}\end{cases}\)

Then, f is

A. one - one and into 

B. one - one and onto 

C. many - one and into 

D. many - one and onto

Answer 1

Correct Answer is (D) many - one and onto.

f(1) = 1 

f(2) = 1 

f(3) = 2 

f(4) = 2 

f(5) = 3 

f(6) = 3 

Since at different values of x we get same value of y ∴f(n) is many –one 

And range of f(n) = N = N(codomain) 

∴ the function f: N → Z, defined by

\(f:N→N:f(x)=\begin{cases}\frac{1}{2}(n+1), \text{ when n is odd}\\ \frac{n}{2},\text{when n is even}\end{cases}\)

is both many - one and onto.