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  4. Number of integral values in the domain of function f(x)= sin -1|(x2-4/3 x)|, is

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Q. Number of integral values in the domain of function $f(x)=\sin ^{-1}\left|\frac{x^2-4}{3 x}\right|$, is

Inverse Trigonometric Functions

Solution:

We must have, $-1 \leq\left|\frac{x^2-4}{3 x}\right| \leq 1 $
$\Rightarrow-1 \leq \frac{x^2-4}{3 x} \leq 1$
$\Rightarrow \frac{x^2-4}{3 x}+1 \geq 0 \text { and } \frac{x^2-4}{3 x}-1 \leq 0 $
$\Rightarrow \frac{(x+4)(x-1)}{3 x} \geq 0 \text { and } \frac{(x-4)(x+1)}{3 x} \leq 0$
$\Rightarrow x =-4,-3,-2,-1,1,2,3,4$.
Hence, number of integral values is 8 .