Examine whether the function f : R → R is one-one(injective) if f ( x ) = x 3 , x ∈ R

Given that function f : R → R is defined by f x = x 3 , x ∈ R We need to check whether this function is one-one or not. Let x 1 , x 2 ∈ R then f x 1 = f x 2 ⇒ x 1 3 = x 2 3 ⇒ x 1 3 - x 2 3 = 0 ⇒ x 1 - x 2 x 1 2 + x 1 x 2 + x 2 2 = 0 Since we know that for x 1 , x 2 ∈ R x 1 2 + x 1 x 2 + x 2 = 0 is not true. Thus, we have x 1 - x 2 = 0 ⇒ x 1 = x 2 Hence, the given function is one-one.

Examine whether the function f : R → R is one-one(injective) if f ( x ) = x 3 , x ∈ R

Given that function f : R → R is defined by f x = x 3 , x ∈ R We need to check whether this function is one-one or not. Let x 1 , x 2 ∈ R then f x 1 = f x 2 ⇒ x 1 3 = x 2 3 ⇒ x 1 3 - x 2 3 = 0 ⇒ x 1 - x 2 x 1 2 + x 1 x 2 + x 2 2 = 0 Since we know that for x 1 , x 2 ∈ R x 1 2 + x 1 x 2 + x 2 = 0 is not true. Thus, we have x 1 - x 2 = 0 ⇒ x 1 = x 2 Hence, the given function is one-one.

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.