Functions 2 Question 9

10.

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

(a) $f(x)=\sin ^{2} x, g(x)=\sqrt{x}$

(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

(1998, 2M)

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Answer:

Correct Answer: 10. (a)

Solution:

  1. 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| $



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