Answer:
{x ∈ ℝ : x ≥ π/6 +2πn and x ≤ π/6 + 2πn and n ∈ ℤ}
Step-by-step explanation:
sinx can run from -1 to +1
2sinx can run from -2 to +2
2sinx -1 can run from -3 to +1
However, the square root is imaginary when x < 0. So, the condition is
2sinx -1 ≥ 0
2sinx ≥ 1
sinx ≥ ½
x ≥ π/6 (30°)
So, in the interval [0, 2π], π/6 ≤ x ≤ 5π/6
However, the sine is a cyclic function and repeats itself every 2π.
Over all real numbers, the condition is (π/6 +2πn) ≤ x ≤ (5π/6 + 2πn).
The domain is then
{x ∈ ℝ : x ≥ π/6 +2πn and x ≤ π/6 + 2πn and n ∈ ℤ}
