Let f: R → R be a function defined by f(x) = (2(1 - x ^ 25 2 )(2 +...

Given $f\left(x\right) = \left[2\left(1 - \dfrac{{x}^{25}}{2}\right)\left(2 + {x}^{25}\right)\right]\dfrac{1}{50}$

$f\left(x\right) = \left[\left(2 - {x}^{25}\right)\left(2 + {x}^{25}\right)\right]\dfrac{1}{50}$

$= \left(4 - {x}^{50}\right)\dfrac{1}{50}$

i.e. $f\left(f\left(x\right)\right) = \left(4 - {\left(\left(4 - {x}^{50}\right)\dfrac{1}{50}\right)}^{50}\right)\dfrac{1}{50} = x$

Now $g\left(x\right) = f\left(f\left(f\left(x\right)\right)\right) + f\left(f\left(x\right)\right)$

$= f\left(x\right) + x$

So $g\left(1\right) = f\left(1\right) + 1 = 3\dfrac{1}{50} + 1$

Hence \left[g\left(1\right)\right] = \left[3\dfrac{1}{50} + 1\right] = 2$