Which one of the following is correct in respect of f(x) = 1/√(|x| - x) and g(x) = 1/√(x - |x|) ?

For f(x) = 1/√(|x| - x): we need |x| - x > 0. If x ≥ 0, |x| - x = 0 (not > 0). If x < 0, |x| - x = -x - x = -2x > 0. So f is defined for x < 0, i.e., f has domain (-∞, 0). For g(x) = 1/√(x - |x|): we need x - |x| > 0. If x ≥ 0, x - |x| = 0. If x < 0, x - |x| = x - (-x) = 2x < 0. So g(x) has no real domain. Therefore f has some domain and g has no domain.