Question

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

f(k) = `{{:(k + 1, if k  "is odd"),(     k, if k  "is even"):}`.

Then the number of possible functions g : A `rightarrow` A such that gof = f is ______.

Options

• 5⁵

• 10⁵

• 5!

• ¹⁰C₅

Answer 1

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

f(k) = `{{:(k + 1, if k  "is odd"),(     k, if k  "is even"):}`.

Then the number of possible functions g : A `rightarrow` A such that gof = f is `underlinebb(10^5)`.

Explanation:

Putting value of K from 1 to 10, we get

f(1) = f(2) = 2

f(3) = f(4) = 4

f(5) = f(6) = 6

f(7) = f(8) = 8

f(9) = f(10) = 10

Since, g(f(x)) = f(x)

∴ gof(1) = f(1) `\implies` g(2) = f(1) = 2

gof(2) = f(2) `\implies` g(2) = f(2) = 2

gof(3) = f(3) `\implies` g(4) = f(3) = 4

∴ The image of 2, 4, 6, 8, 10 in function g(x) should be 2, 4, 6, 8, 10 respectively. Therefore, image of each of remaining elements can be any of 10 elements.

Hence, number of possible g(x) is 10⁵.