Question

A function f: $N^{+}\to N^{+}$, defined on the set of positive integers $N^{+}$, satisfies the following properties

f(n) = f$\left(\frac{n}{2}\right)$ if n is even

f(n) = f(n + 5) if n is odd

Let R = $\{i\mid \exists j:f(j=i\left)\right\}$ be the set of distinct value that f takes. The maximum possible size of R is

• A. 2

Answer 1

The correct option is A 2
Given that

f(n) = f$\left(\frac{n}{2}\right)$ if n is even

f(n) = f(n + 5) if n is odd

We can observe that

Case I: f(1) = f(6) = f(3) = f(8) = f(2) = f(1)..........

Case II : f(5) = f(10) = f(5) ..........

Now, f(7) = f(12) = f(6)

and f(9) = f(14) = f(7) = f(12) = f(6)

So f(7) and f(9) are not distinct, they can be obtained by using case I.

So overall, range of f(x) will contain 2 distinct elements only, f(1) and f(5).