SATHEE: Functions 2 Question 9

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) $f (x) = \sin x, g (x) = | x |$

(d) $f$ and $g$ cannot be determined

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Answer:

Correct Answer: 10. (a)

Solution:

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

Now, $f \circ g (x) = f [g (x)] = f (\sqrt{x}) = \sin^{2} \sqrt{x}$

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

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

$(g \circ f) (x) = g [f (x)] = g (x^{2}) = \sin \sqrt{x^{2}}$

$= \sin | x | \neq | \sin x |$

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Functions 2 Question 9

10. If gf(x)=|sin⁡x| and fg(x)=(sin⁡x)2, then

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