First slide

Functions (XII)

Question

$If f (x) = \frac{x}{\sqrt{(1 + x^{2})}}, then (fofof) (x) =$

Easy

A

$\frac{3 x}{\sqrt{(1 + x^{2})}}$
B

$\frac{x}{\sqrt{(1 + 3 x^{2})}}$
C

$\frac{3 x}{\sqrt{(1 - x^{2})}}$
D

None of these

Solution

$f (x) = \frac{x}{\sqrt{1 + x^{2}}} \Rightarrow f (f (x)) = f (\frac{x}{\sqrt{1 + x^{2}}}) = \frac{\frac{x}{\sqrt{1 + x^{2}}}}{\sqrt{1 + \frac{x^{2}}{1 + x^{2}}}} = \frac{x}{\sqrt{2 x^{2} + 1}}$
$\Rightarrow f (f (f (x))) = f (\frac{x}{\sqrt{2 x^{2} + 1}}) = \frac{\frac{x}{\sqrt{2 x^{2} + 1}}}{\sqrt{1 + \frac{x^{2}}{2 x^{2} + 1}}} = \frac{x}{\sqrt{3 x^{2} + 1}}$

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