If g(f(x))=| | and f(g(x))=( x )^2, then - Vidyadip

MCQ

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

✓
$\text{f(x)}=\sin^2\text{x},\ \text{g(x)}=\sqrt{\text{x}}$
B
$\text{f(x)}=\sin\text{x},\ \text{g(x)}=|\text{x}|$
C
$\text{f(x)}=\text{x}^2,\ \text{g(x)}=\sin\sqrt{\text{x}}$
D
$\text{f and g cannot be determined.}$

✓

Answer

Correct option: A.

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

If we solve it by the trial$-$and$-$error method, we can see that $(a)$ satisfies the given condition.
From $(a):$
$\text{f(x)}=\sin^2\text{x}$ and $\text{g(x)}=\sqrt{\text{x}}$
$\Rightarrow\ \text{f(g(x))}=\text{f}(\sqrt{\text{x}})=\sin^2\sqrt{\text{x}}$
$=(\sin\sqrt{\text{x}})^2$

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