If f x = 2 + 2 x , - 1 ≤ x < 0 1 - x 3 , 0 ≤ x ≤ 3 ; g x = - x , - 3 ≤ x ≤ 0 x , 0 < x ≤ 1 , then range of ( f ∘ g ( x ) ) is

Step-by-step solution

Given: f x = 2 + 2 x , x ∈ [ - 1 , 0 ) 1 - x 3 , x ∈ 0 , 3 and g x = x , x ∈ ( 0 , 1 ] - x , x ∈ - 3 , 0 ⇒ f g x = 2 + 2 g x , g x ∈ [ - 1 , 0 ) 1 - g x 3 , g x ∈ 0 , 3 ⇒ f g x = 2 + 2 x , x ∈ [ - 1 , 0 ) ⇒ x ∈ ϕ 1 - x 3 , x ∈ 0 , 3 as x ∈ - 3 , 3 for x ≤ 3 but x ∈ - 3 , 1 in g x ⇒ f g x = 1 - x 3 , x ∈ - 3 , 1 Now, maximum value of f g x = 1 - x 3 = 1 - 0 3 = 1 And minimum value is given by f g x = 1 - 3 3 = 1 - 1 = 0 Hence, the range of the function will be 0 , 1