Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) 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 \(g(x)=f(f(f(x)))+f(f(x))\), then the greatest integer less than or equal to \(\mathrm{g}(1)\) is ____

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Relations And Functions XII

Let

f : R \to 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: 2

First, simplify

f (x)

f (x) = [2 (1 - \frac{x ^{25}}{2}) (2 + x^{25})]^{\frac{1}{50}}

= [(2 - x^{25}) (2 + x^{25})]^{\frac{1}{50}}

= [4 - x^{50}]^{\frac{1}{50}}

Now compute

f (f (x))

f (f (x)) = [4 - ((4 - x^{50})^{\frac{1}{50}})^{50}]^{\frac{1}{50}}

= [4 - (4 - x^{50})]^{\frac{1}{50}}

= [x^{50}]^{\frac{1}{50}} = x

Therefore:

g (x) = f (f (f (x))) + f (f (x)) = f (x) + x

x = 1

g (1) = f (1) + 1 = (4 - 1)^{\frac{1}{50}} + 1 = 3^{\frac{1}{50}} + 1

Since

3^{\frac{1}{50}} \approx 1.022

, we have

g (1) \approx 2.022

.

The greatest integer less than or equal to

g (1)

is 2.

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Integer answer