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

Correct Answer: 2

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

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

\(=(4−x^{50})^{\frac {1}{50}}\)
\(f(f(x))=(4−((4−x^{50})^{\frac {1}{50}})^{50})^{\frac {1}{50}}=x\)
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 \(2\).