Q87 / 200 · Manipal (MET) 2014 · Mathematics

If $f \circ g=|\sin x|$ and $g \circ f=\sin ^2 \sqrt{x}$, then $f(x)$ and $g(x)$ are

  • A
    $f(x)=\sqrt{\sin x}, g(x)=x^2$
  • B
    $f(x)=|x|, g(x)=\sin x$
  • Correct answer
    C
    $f(x)=\sqrt{x}, g(x)=\sin ^2 x$
  • D
    $f(x)=\sin \sqrt{x}, g(x)=x^2$

Solution

$\begin{aligned} & f \circ g=f\{g(x)\}=|\sin x|=\sqrt{\sin ^2 x} \\ & \text { Also, } g \circ f=g\{f(x)\}=\sin ^2 \sqrt{x} \\ & \text { Obviously, } \quad \sqrt{\sin ^2 x}=\sqrt{g(x)} \\ & \text { and } \quad \sin ^2 \sqrt{x}=\sin ^2\{f(x)\} \\ & \text { i.e., } \quad g(x)=\sin ^2 x \\ & \text { and } \quad f(x)=\sqrt{x} \end{aligned}$

Hence, the correct answer is option: 'C'.

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