Question

Let A = {1, 2, 3,.....,10} and f : A → A be defined as 

Then the number of possible functions g : A → A such that go f = f is

(1) 10⁵

(2) ¹⁰C₅

(3) 5⁵

(4) 5!

Answer 1

Correct option is (1) 10⁵

\(\therefore\) g : A → A such that g(f(x)) = f(x)

⇒ If x is even then g(x) = x .....(1)

If x is odd then g(x + 1) = x + 1 ...(2)

from (1) and (2) we can say that

g(x) = x if x is even

⇒ If x is odd then g(x) can take any value in set A

so number of g(x) = 10⁵ × 1