1. Tardigrade
  2. Question
  3. Mathematics
  4. Consider, f(x)=(|x-4|/|x|+1). If sum of all distinct possible values of sin -1( sin [f(x)]) is a π+b then find the absolute value of (a+b). [Note: [z] denotes greatest integer function less than or equal to z.]

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Q. Consider, $f(x)=\frac{|x-4|}{|x|+1}$. If sum of all distinct possible values of $\sin ^{-1}(\sin [f(x)])$ is $a \pi+b$ then find the absolute value of $(a+b)$.
[Note: [z] denotes greatest integer function less than or equal to $z$.]

Inverse Trigonometric Functions

Solution:

Range of $f ( x )$ is $[0,4]$
$\therefore \sin ^{-1}(\sin [ f ( x )])=\sin ^{-1}(\sin 0)+\sin ^{-1}(\sin 1)+\sin ^{-1}(\sin 2)+\sin ^{-1}(\sin 3)+\sin ^{-1}(\sin 4) $
$ =0+1+\pi-2+\pi-3+\pi-4=3 \pi-8 \equiv a \pi+ b $
$\therefore ( a + b )=|3-8|=5 $