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← JEE Main 2022 · 25 Jun · Shift 1
MathematicsInteger+4 / -1Question 27 of 90

JEE Main 2022 · 25 Jun · Shift 1, Mathematics question 27

Let f:R→Rf:\mathbb R\to\mathbb R be a function defined by f(x)=(2(1−x252)(2+x25))150f(x)=\left(2\left(1-\dfrac{x^{25}}2\right)\left(2+x^{25}\right)\right)^{\frac1{50}}. If the function g(x)=f(f(f(x)))+f(f(x))g(x)=f(f(f(x)))+f(f(x)), then the greatest integer less than or equal to g(1)g(1) is ______.

Answer and worked solution

Official answer: 2

Simplify. f(x)=((2−x25)(2+x25))1/50=(4−x50)1/50f(x)=\left((2-x^{25})(2+x^{25})\right)^{1/50}=\left(4-x^{50}\right)^{1/50}.

Compose. f(f(x))=(x50)1/50=∣x∣f(f(x))=\left(x^{50}\right)^{1/50}=|x|, so f(f(f(x)))=f(∣x∣)f(f(f(x)))=f(|x|).

At 1. g(1)=f(1)+1=31/50+1≈2.02g(1)=f(1)+1=3^{1/50}+1\approx2.02; the greatest integer is 2.

Answer: 2

Topics:Composition and inverse functions
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