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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
Solution
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
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