Grade 12IIT JEE Entrance Exam

Find the domain of the functions given in the attached image.

Question image for Find the domain of the functions given in the att

Vishnu MK

10 Years agoGrade 12

1 Answer

Riddhish Bhalodia

10 Years ago

1.
The first square root requires $x \leq 0$ but if x=0 then the second term denominator goes to zero and hence the domain is
$x < 0$

2.
The denominator must not be zero and the whole rational function must be positive
when x<0,|x| = -x and then if x<-1 then both numerator and denominator becomes neative and hence making the rational function positive, secondly for 1>x>=0 the denominator is positive which works too
Hence the domain is
$x \in (-\infty,-1) \cup [0,1)$

3.
we deal with quadratic inside square roots
first one will give
$x^2-|x| -2 = (|x|-2)(|x|+1) \geq 0$
|x| is always positive therefore what remains is
$(|x|-2)\geq 0 \Rightarrow x \in (-\infty,-2]\cup[2,\infty)$
Now the second quadratic (taking the -ve)
$x^2 - 16 = (x-4)(x+4) \leq 0$
$\Rightarrow x \in [-4,4]$
taking the intersection of both the subdomains we get the domain as
$x \in [-4,-2]\cup[2,4]$