Q78 / 160 · TS EAMCET 2012 · Mathematics

If $f: R \rightarrow R^2$ and $g: R^{+} \rightarrow R$ are such that $g\{f(x)\}=|\sin x| \quad$ and $\quad f\{g(x)\}=(\sin \sqrt{x})^2$, then a possible choice for $f$ and $g$ is

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

Solution

Given, $g\{f(x)\}=|\sin x|$ and $\quad f\{g(x)\}=(\sin \sqrt{x})^2$ Let us consider $f(x)=\sin ^2 x$ and $g(x)=\sqrt{x}$ $\therefore \quad f\{g(x)\}=f(\sqrt{x})=\left(\sin ^2 \sqrt{x}\right)=(\sin \sqrt{x})^2$ and $g\{f(x)\}=g\left(\sin ^2 x\right)=\sqrt{\sin ^2 x}=|\sin x|$

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

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