Question

If \( f \circ g(x)=1-x, \forall x \in R \) and \( f \circ f(x)=1+\sin (f(x)), \forall x \in R \), then 

a) \( f(0)=1 \) 

b) \( f(\pi)=1 \) 

c) \( f(x) \) is periodic 

d) \( f(x) \geq 0, \forall x \in \)

Answer 1

\(f\ o \ f(x)\) = 1 + sin (f(x))

Let x = f^-1(y)

Then \(f\ o \ f(f^{-1}(y))\) = 1 + sin(f(f^-1(y))

⇒ f(x) = 1+ sin(y) (∵ f(f^-1(y)) = y)

⇒ f(x) = 1+ sin x

∴ f(0) = 1,

f(π) = 1,

f(x) is periodic with period 2π

f(x) >,0 ∀ x ∈ (∵ sin x ∈[-1.1])

All four options are correct.