The domain of the real function f(x) = ⁻¹ ( ₂ ( x^2 2 ) ) is

Step-by-step solution

For the function f(x) = ⁻¹ ( ₂ ( x^2 2 ) ) to be defined, the argument of the inverse sine function must lie in the interval [-1, 1] . -1 ₂ ( x^2 2 ) 1 Using the property of logarithms, we get: 2⁻¹ x^2 2 2^1 1 2 x^2 2 2 Multiplying the entire inequality by 2 , we obtain: 1 x^2 4 Taking the square root yields: 1 |x| 2 This implies that x must satisfy either -2 x -1 or 1 x 2 . Thus, the domain of the function is x [-2, -1] [1, 2] . Answer: [-2, -1] [1, 2]