NDA 1 2024 Maths, question 80

Topic: Differential calculus

Q80

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 domainCorrect answer
(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

Solution.

Answer: (a) $f(x)$ has some domain and $g(x)$ has no domain

For f, the radicand must be strictly positive because it is in a denominator.
When x < 0, |x| − x = −2x > 0, so its domain is (−∞, 0).
But x − |x| is never positive, so g has no real domain.