If f(f(x)) = x for all x in domain, and f is one-one, then f(f(x)) = x. (Find the property)

Solution

Maths sir · 3 steps

Read f(f(x)) = x as a statement about the composite map: f∘f is the identity, so f undoes itself.

1. Composite map. The condition f(f(x)) = x says that f ° f sends every input back to itself. That is the definition of the identity function I(x) = x.
2. Self-inverse. Since f ° f = I, applying f a second time cancels the first. So f is its own inverse, f⁻¹ = f, and such a function is called an involution. It is automatically one-one and onto, because it has an inverse.

   f ° f = I ⇒ f(f(x))/x = 1 (x ≠ 0)

3. So. The composite f ° f changes nothing, like multiplying by 1. The identity returns its input unchanged, for example f(f(1)) = 1. The required value is 1.

The idea to keep. Whenever you see f(f(x)) = x, think "f is its own inverse". The composite is the identity map, so it returns every input unchanged. Examples are f(x) = -x, f(x) = 1/x and f(x) = a - x.

Answer from the NTA final answer key. Solution by PrepWiser.