Let f x = sin ⁡ π 6 sin ⁡ π 2 sin ⁡ x , for all x ∈ R and g x = π 2 sin ⁡ x , for all x ∈ R . Let f o g x denote f g x and g o f x denote g f x . Then which of the following is (are) true?

Question

Let f x = sin &#8289; &#960; 6 sin &#8289; &#960; 2 sin &#8289; x , for all x &#8712; R and g x = &#960; 2 sin &#8289; x , for all x &#8712; R . Let f o g x denote f g x and g o f x denote g f x . Then which of the following is (are) true?

Quantrex Academy · Accepted Answer

Correct answer: A. Range of f is - 1 2 , 1 2

f x = sin ⁡ π 6 sin ⁡ π 2 sin ⁡ x - π 2 ≤ π 2 sin ⁡ x ≤ π 2 - 1 ≤ sin ⁡ π 2 sin ⁡ x ≤ 1 sin ⁡ - π 6 ≤ sin ⁡ π 6 sin ⁡ π 2 sin ⁡ x ≤ sin ⁡ π 6 - 1 2 ≤ sin ⁡ π 6 sin ⁡ π 2 sin ⁡ x ≤ 1 2 f g x = sin ⁡ π 6 sin ⁡ π 2 sin ⁡ π 2 sin ⁡ x - 1 ≤ sin ⁡ π 2 sin ⁡ x ≤ 1 - π 2 ≤ π 2 sin ⁡ π 2 sin ⁡ x ≤ π 2 - 1 ≤ sin ⁡ π 2 sin ⁡ π 2 sin ⁡ x ≤ 1 - π 6 ≤ π 6 sin ⁡ π 2 sin ⁡ π 2 sin ⁡ x ≤ π 6 - 1 2 ≤ sin ⁡ π 6 sin ⁡ π 2 sin ⁡ π 2 sin ⁡ x ≤ 1 2 lim x → 0 f x g x = lim x → 0 ⁡ sin ⁡ π 6 sin π 2 sin x π 2 sin ⁡ x × π 6 sin ⁡ π 2 sin ⁡

Let f x = sin ⁡ π 6 sin ⁡ π 2 sin ⁡ x , for all x ∈ R and g x = π 2 sin ⁡ x , for all x ∈ R . Let f o g x denote f g x and g o f x denote g f x . Then which of the following is (are) true?

Options

Correct answer

Step-by-step solution

More from Functions

Let f x = sin &#8289; &#960; 6 sin &#8289; &#960; 2 sin &#8289; x , for all x &#8712; R and g x = &#960; 2 sin &#8289; x , for all x &#8712; R . Let f o g x denote f g x and g o f x denote g f x . Then which of the following is (are) true?

Options

Correct answer

Step-by-step solution

More from Functions

Let f x = sin ⁡ π 6 sin ⁡ π 2 sin ⁡ x , for all x ∈ R and g x = π 2 sin ⁡ x , for all x ∈ R . Let f o g x denote f g x and g o f x denote g f x . Then which of the following is (are) true?