Question

$Iff\left(x\right)=\frac{x}{\sqrt{1+x^2}}$, then$\left(fofof\right)\left(x\right)=$

• A. $\frac{3x}{\sqrt{1+x^2}}$
• B. $\frac{x}{\sqrt{1+3x^2}}$
• C. $\frac{3x}{\sqrt{1+x^2}}$
• D. None of these

Answer 1

The correct option is B $\frac{x}{\sqrt{1+3x^2}}$

$\left(fofofo\right)\left(x\right)=\left(fof\right)\left(f\right(x\left)\right)=\left(fof\right)\left(\frac{x}{\sqrt{1+x^2}}\right)$

$=f\left[\frac{\left(\frac{x}{\sqrt{1+x^2}}\right)}{\sqrt{1+\frac{x^2}{1+x^2}}}\right]=f\left(\frac{x\sqrt{1+x^2}}{\sqrt{1+x^2}\sqrt{1+2x^2}}\right)$

$=f\left(\frac{x}{\sqrt{1+2x^2}}\right)=\frac{\frac{x}{\sqrt{1+2x^2}}}{\sqrt{[1+\frac{x^2}{1+2x^2}]}}=\frac{x}{\sqrt{1+3x^2}}.$