If f : R → R such that f ( x ) = 3 − x 3 1 3 , then prove that f ∘ f x = x .

We have, f ( x ) = 3 − x 3 1 3 Consider, f ( f ( x ) ) = 3 − f ( x ) 3 1 3 ⇒ f ∘ f ( x ) = 3 − 3 − x 3 1 3 × 3 1 3 ⇒ f ∘ f ( x ) = 3 − 3 − x 3 1 3 ⇒ f ∘ f ( x ) = 3 − 3 + x 3 1 3 ⇒ f ∘ f ( x ) = x 3 1 3 ⇒ f ∘ f ( x ) = x Thus f ∘ f ( x ) = x Hence, proved.

Find g o f and f o g if f x = 8 x 3 and g x = x 1 3 .

Given, f x = 8 x 3 and g x = x 1 3 For g o f x , g o f x = g f x = g 8 x 3 [ ∵ Applying the definition of f x .] = 8 x 3 1 3 [ ∵ Applying the definition of g x .] = 2 x 3 1 3 ⇒ g o f x = 2 x For f o g x , f o g x = f g x = f x 1 3 [ ∵ Applying the definition of g x .] = 8 x 1 3 3 [ ∵ Applying the definition of f x .] ⇒ f o g x = 8 x

If f : R → R is defined by f x = x 2 - 3 x + 2 , find f f x .

Given f : R → R is defined by f x = x 2 - 3 x + 2 Now f f ( x ) = f x 2 - 3 x + 2 ⇒ f f x = x 2 - 3 x + 2 2 - 3 x 2 - 3 x + 2 + 2 ⇒ f f x = x 4 + 9 x 2 + 4 - 6 x 3 - 12 x + 4 - 3 x 2 + 9 x - 6 + 2 a + b + c 2 = a 2 + b 2 + c 2 + 2 a b + 2 b c + 2 c a ⇒ f f x = x 4 - 6 x 3 + 10 x 2 - 3 x .