Question

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

Answer 1

$f(x)=(2(1−\frac {x^{25}}{2})(2+x^{25}))^{\frac {1}{50}}$

$f(x)=(2(\frac {2−x^{25}}{{2}})(2+x^{25}))^{\frac {1}{50}}$

$=(4−x^{50})^{\frac {1}{50}}$
$f(f(x))=(4−((4−x^{50})^{\frac {1}{50}})^{50})^{\frac {1}{50}}=x$
As $f(f(x)) = x$ we have
$g(x) = f(f(f(x))) + f(f(x)) = f(x) + x$
$g(x) = (4 – x^{50})^{\frac {1}{50}} + x$
$g(1) = 3^{\frac {1}{50}} \+ 1$
$[g(1)] = 2$

So, the answer is $2$.