Question

Let f: [0, 1] → [0, 1] be defined by f(x) = `{{:(x",",  "if"  x  "is rational"),(1 - x",",  "if"  x  "is irrational"):}`. Then (f o f) x is ______.

Options

• Constant

• 1 + x

• x

• None of these

Answer 1

Let f: [0, 1] → [0, 1] be defined by f(x) = `{{:(x",",  "if"  x  "is rational"),(1 - x",",  "if"  x  "is irrational"):}`. Then (f o f) x is x.

Explanation:

Given that, f: [0, 1] → [0, 1] be defined by

f(f(x)) = `{{:(f(x)",",  "if"  f(x)  "is rational"),(1 - f(x)",",  "if"  f(x)  "is  irrational"):}`

=  `{{:(x",",  "if"  x  "is rational"),(1 - (1 - x)",",  "if"  1 - x  "is  irrational"):}`

= `{{:(x",",  "if"  x  "is rational"),(x",",  "if"  x  "is irrational"):}`

∴ (fof)x = f(f(x)) = x