If f x = cos ⁡ x + sin ⁡ x and g x = x 2 - 1 , then g f x is injective in the interval

Question

If f x = cos ⁡ x + sin ⁡ x and g x = x 2 - 1 , then g f x is injective in the interval, 0 , π 2, – π 4 , π 4, – π 2 , π 2, 0 , π

MARKS App · Accepted Answer

g f x = sin ⁡ x + cos ⁡ x 2 - 1 = sin ⁡ 2 x Now, sin ⁡ 2 x is injective if 2 x ∈ n π - π 2 , n π + π 2 ⇒ x ∈ n π 2 - π 4 , n π 2 + π 4 ∀ n ∈ Z