Find the domain and range of the function .

We have ,

Domain:

We know that,

-1 ≤ sin 3x ≤ 1 for all x ∈ R

⇒ -1 ≤ - sin 3x ≤ 1 for all x ∈ R

⇒ -1+2 ≤ 2 - sin 3x ≤ 2+1 for all x ∈ R

⇒ 1 ≤ 2 - sin 3x ≤ 3 for all x ∈ R

⇒ 2 – sin 3x ≠ 0 for any x ∈ R

Hence domain is R.

Range:

As 1 ≤ 2 - sin 3x ≤ 3 for all x ∈ R

for all x ∈ R

for all x ∈ R

⇒ f(x) ∈ [ 1/3,1]

Hence range is [ 1/3,1].