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# Let $f:R \rightarrow R$ be defined by $f(x)=\Large {\frac{x}{\sqrt{1+x^2}}}$.Then ( f o f o f )(x)=____________.

Toolbox:
• $(fofof)x=f(f(f(x)))$
$f:R \to R$

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

$fof=f\bigg(\large\frac{x}{\sqrt {1+x^2}}\bigg)$

$=\large\frac{\large\frac{x}{\sqrt {1+x^2}}}{\sqrt {1+\bigg(\frac{x}{\sqrt{1+x^2}}\bigg)^2}}$

$=\large\frac{\large\frac{x}{\sqrt {1+x^2}}}{\frac{\sqrt {1+x^2+x^2}}{\sqrt {1+x^2}}}$

$=\frac{x}{\sqrt {1+2x^2}}$

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

$=\large\frac{\large\frac{x}{\sqrt {1+2x^2}}}{\sqrt {1+\bigg(\frac{x}{\sqrt{1+2x^2}}\bigg)^2}}$

$=\large\frac{\large\frac{x}{\sqrt {1+2x^2}}}{\frac{\sqrt {1+2x^2+x^2}}{\sqrt {1+2x^2}}}$

$=\frac{x}{\sqrt {1+3x^2}}$

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