Functions 2 Question 9
10. If $g{f(x)}=|\sin x|$ and $f{g(x)}=(\sin \sqrt{x})^{2}$, then
(a) $f(x)=\sin ^{2} x, g(x)=\sqrt{x}$
(1998, 2M)
(b) $f(x)=\sin x, g(x)=|x|$
(c) $f(x)=x^{2}, g(x)=\sin \sqrt{x}$
(d) $f$ and $g$ cannot be determined
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Answer:
Correct Answer: 10. (a)
Solution:
- Let
$$ f(x)=\sin ^{2} x \text { and } g(x)=\sqrt{x} $$
Now, $\quad f \circ g(x)=f[g(x)]=f(\sqrt{x})=\sin ^{2} \sqrt{x}$
and $\quad g \circ f(x)=g[f(x)]=g\left(\sin ^{2} x\right)=\sqrt{\sin ^{2} x}=|\sin x|$
Again, let $f(x)=\sin x, g(x)=|x|$
$$ \begin{aligned} f \circ g(x) & =f[g(x)]=f(|x|) \\ & =\sin |x| \neq(\sin \sqrt{x})^{2} \end{aligned} $$
When
$$ f(x)=x^{2}, g(x)=\sin \sqrt{x} $$
$$ f \circ g(x)=f[g(x)]=f(\sin \sqrt{x})=(\sin \sqrt{x})^{2} $$
and
$$ (g \circ f)(x)=g[f(x)]=g\left(x^{2}\right)=\sin \sqrt{x^{2}} $$
$$ =\sin |x| \neq|\sin x| $$
