Question

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$\large\bold\red{Q.}$ For a real number $\bold{x}$ ,
Let f(x) = 0 if x < 1 and
f(x) = 2x - 2 if x ≥ 1.

If number of solutions to the Equation ,
f(f(f(f(x)))) = x is n ,

Then find the value of $\dfrac{n}{5}$.

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Answer 1

Given

F(X) = 0 if F(X) <1

F(X) = 2x - 2 where x> 1

Now

F(X) = 2x - 2

F(FX) = 2(2x - 2) - 2 = 4x -4

F(F(F(X))) = 4(2x - 2) - 4

F(F(F(X)))= 8x - 12

F(F(F(F(F(X) = 8(2x - 2) - 4

F(F(F(F(F(X) = 16x - 20 = X

15x - 20 = 0

5(3x - 4) = 0

5(sqrt3x - 2)(sqrt3 + 2) = 0

number of solutions is 2

so n = 2

n/5 = 2/5

Answer 2

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