The domain set of definition of the function $f(x)=\sqrt{\cos(\sin x)}+\sin^{-1}\left(\frac{1+x^2}{2x}\right)$ is
Answer & explanation
Correct answer: option 4
cos (sin x) ≥ 0 for all x ; x ≠ 0
Hence cos (sin x) > 0 for all x
$-\frac{π}{2}≤\sin^{-1}\left(\frac{1+x^2}{2x}\right)≤+\frac{π}{2}$
i.e. $-1≤\frac{1+x^2}{2x}⇒-2x≥(1+x^2)⇒(1+x)^2≤0⇒x=-1$
The other side $(x>0),1+x^2≤2x⇒(x-1)^2≤0⇒x=1$
The function is defined only for the two values x = 1 and x = −1.