Question

Let $f : R \rightarrow R$ be a function defined by $f(x)=\left(2\left(1-\frac{x^{25}}{2}\right)\left(2+x^{25}\right)\right)^{\frac{1}{50}}$. If the function $g(x)=f(f(f(x)))+f(f(x))$, the the greatest integer less than or equal to $g(1)$ is ______

Answer 1

$f(x)=\left[2\left(1-\frac{x^{25}}{2}\right)\left(2+x^{25}\right)\right]^{\frac{1}{50}}$
$f(x)=\left[\left(2-x^{25}\right)\left(2+x^{25}\right)\right]^{\frac{1}{50}}$
$=\left(4-x^{50}\right)^{1 / 50}$
$f(f(x))=\left(4-\left(\left(4-x^{50}\right)^{1 / 50}\right)^{50}\right)^{1 / 50}=x$
$g(x)=f(f(f(x)))+f(f(x))$
$=f(x)+x$
$g(1)=f(1)+1=3^{1 / 50}+1$
$[g(1)]=\left[3^{1 / 50}+1\right]=2$