Which one of the following is correct in respect of f(x)=\frac{1}{\sqrt{|x|-x}} and g(x)=\frac{1}{\sqrt{x-|x|}}?
- A. f(x) has some domain and g(x) has no domain ✓
- B. f(x) has no domain and g(x) has some domain
- C. f(x) and g(x) have the same domain
- D. f(x) and g(x) do not have any domain
Correct Answer: A. f(x) has some domain and g(x) has no domain
Explanation
For f(x), we need |x| - x \gt 0 \implies |x| \gt x, which holds for x \lt 0. So f(x) has domain (-\infty, 0). For g(x), we need x - |x| \gt 0 \implies x \gt |x|, which is false for all real x. So g(x) has no real domain.
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