If g f x -sin x and fg x -sin x 2- thena f x-sin 2 x- g x-xb fx -sin x - g x - xc fx - x 2- g x -sin xd f and g cannot be determined-

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Byju's Answer

Standard XII

Mathematics

Continuity in an Interval

If g f x =sin...

Question

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.

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Solution

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

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If g f x=sin x and f g x=sin x2, then (a) fx=sin2 x, gx=x (b) fx=sin x, gx=|x| (c) fx=x2, gx=sin x (d) f and g cannot be determined.

If we solve it by the trial-and-error method, we can see that (a) satisfies the given condition. From (a): fx=sin2x and gx=x⇒fgx=fx=sin2 x=sin x2 So, the answer is (a).

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.

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