Let f:R → R be a function defined by
f(x) = ( 2( 1 - x²⁵/2 )(2 + x²⁵) )^(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 ____________.

Solution

Maths sir · 4 steps

Multiply out the brackets first: f(x) becomes (4 - x⁵⁰)^(1/50), and applying f twice undoes it.

1. Tidy up f. 2(1 - x²⁵/2) = 2 - x²⁵, so the product is (2 - x²⁵)(2 + x²⁵) = 4 - x⁵⁰. That gives f(x) = (4 - x⁵⁰)^(1/50).
2. Apply f twice. Put f(x) back into f. Raising f(x) to the power 50 gives back 4 - x⁵⁰, so the 4 cancels.

   f(f(x)) = (4 - (4 - x⁵⁰))^(1/50) = (x⁵⁰)^(1/50) = |x|

3. Evaluate at 1. f(f(1)) = 1, so f(f(f(1))) = f(1) = 3^(1/50). Then g(1) = 3^(1/50) + 1.
4. So. Since 1 < 3^(1/50) < 2 (it is about 1.022), g(1) lies between 2 and 3. The greatest integer less than or equal to g(1) is 2.

The idea to keep. When f looks heavy, simplify its formula before composing. Here f(x) = (a - xⁿ)^(1/n) with n even, so f(f(x)) = |x|, and three applications collapse to one.

Answer from the NTA final answer key. Solution by PrepWiser.