The domain of the function `f(x)=(sin^(-1)(3-x))/(I n(|x|-2)i s` (a) `[2,4]` (b) `(2,3)uu(3,4]` (c) `(0,1)uu(1,oo)` (d) `(-oo,-3)uu(2,oo)`

Options

A. `[2,4]`
B. `(2,3) cup (3,4]`
C. `[2,oo)`
D. `(-oo,-3) cup [2,oo)`

Answer

B

Explanation

`f(x)=(sin^(-1)(3-x))/(log(|x|-2))`
Let `g(x)=sin^(-1)(3-x)`
or `-1 le 3 -x le 1`
The domain of g(x) is [2, 4].
Let `h(x)=log(|x|-2)`
I.e., `|x|-2 gt 0 " or " |x| gt 2`
i.e., `x lt -2 " or " x gt 2`
` :. " Domain " (-oo, -2) cup (2 ,oo)`
We know that
`(f//g)(x)=(f(x))/(g(x)) AA x in D_(1) cap D_(2)-{x in R:g(x)=0}`
Therefore, the domain of `f(x) " is " (2,4]-{3}=(2,3) cup (3,4].`