Let f x = e x - x and g x = x 2 - x , ∀ x ϵ R . Then the set of all x ϵ R , where the function h x = f o g x is increasing, is:

Step-by-step solution

h x = f o g x = e g x - g x = e x 2 - x - x 2 - x h ' x = e x 2 - x × 2 x - 1 - 2 x + 1 = 2 x - 1 e x 2 - x - 1 ∵ h x is increasing ∴ h ' x ≥ 0 ⇒ 2 x - 1 e x 2 - x - 1 ≥ 0 Case 1: 2 x - 1 ≥ 0 & x 2 - x ≥ 0 ⇒ x ≥ 1 2   &   x ∈ - ∞ ,   0 ∪ 1 , ∞ ⇒ x ∈ 1 ,   ∞ Case 2: 2 x - 1 ≤ 0 and x 2 - x ≤ 0 x < 1 2 and x ∈ 0 ,   1 ⇒ x ∈ 0 , 1 2 ∴   &#160