First slide

Functions (XII)

Question

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

Moderate

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.

Solution

$When f (x) = \sin^{2} x and g (x) = \sqrt{x}, (fog) (x) = f [g (x)] = f (\sqrt{x}) = (\sin \sqrt{x})^{2} and (gof) (x) = g [f (x)] = g (\sin^{2} x) = | \sin x | When f (x) = \sin x and g (x) = | x | (fog) (x) = f (g (x)) = f (| x |) = \sin | x | \neq (\sin \sqrt{x})^{2} When f (x) = x^{2} and g (x) = \sin \sqrt{x} (fog) (x) = f [g (x)] = f (\sin \sqrt{x}) = (\sin \sqrt{x})^{2} and (gof) (x) = g [f (x)] = g (x^{2}) = \sin \sqrt{x^{2}} = \sin | x | \neq | \sin x |$

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