Question

If $g\left(x\right)=\frac{1}{f\left(x\right)}$ and $f\left(x\right)=x,x\ne 0,$ then which one of the following is correct?

• A. $f\left(f\right(f\left(g\right(g\left(f\right(x\left)\right)\left)\right)\left)\right)=g\left(g\right(f\left(g\right(f\left(x\right)\left)\right)\left)\right)$
• B. $f\left(g\right(f\left(g\right(g\left(f\right(g\left(x\right)\left)\right)\left)\right)\left)\right)=g\left(g\right(f\left(g\right(f\left(x\right)\left)\right)\left)\right)$
• C. $f\left(g\right(f\left(g\right(g\left(f\right(g\left(x\right)\left)\right)\left)\right)\left)\right)=f\left(g\right(f\left(g\right(f\left(x\right)\left)\right)\left)\right)$
• D. $f\left(f\right(f\left(g\right(g\left(f\right(x\left)\right)\left)\right)\left)\right)=f\left(f\right(f\left(g\right(f\left(x\right)\left)\right)\left)\right)$

Answer 1

The correct option is C $f\left(g\right(f\left(g\right(g\left(f\right(g\left(x\right)\left)\right)\left)\right)\left)\right)=f\left(g\right(f\left(g\right(f\left(x\right)\left)\right)\left)\right)$

From the given data, it is evident that $gof\left(x\right)=\frac{1}{x}=fog\left(x\right)$

$f\left(f\right(f\left(g\right(g\left(f\right(x\left)\right)\left)\right)\left)\right)=x$ and $g\left(g\right(f\left(g\right(f\left(x\right)\left)\right)\left)\right)=\frac{1}{x}$

$f\left(g\right(f\left(g\right(g\left(f\right(g\left(x\right)\left)\right)\left)\right)\left)\right)=x$ and $g\left(g\right(f\left(g\right(f\left(x\right)\left)\right)\left)\right)=\frac{1}{x}$

$f\left(g\right(f\left(g\right(g\left(f\right(g\left(x\right)\left)\right)\left)\right)\left)\right)=x$ and $f\left(g\right(f\left(g\right(f\left(x\right)\left)\right)\left)\right)=x$

$f\left(f\right(f\left(g\right(g\left(f\right(x\left)\right)\left)\right)\left)\right)=x$ and $f\left(f\right(f\left(g\right(f\left(x\right)\left)\right)\left)\right)=\frac{1}{x}$