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Most Important Selected Qs for JEE AdvancedMathematicsFunctions

If f is continuous function satisfying f(f(x))=1+x x R . then f(1) is equal to

Correct answer

1

Step-by-step solution

Consider g ( x )= f ( x )- x - 1 2 for any x = a aligned & g(a)=f(a)-a- 1 2 & g(f(a))=f(f(a))-f(a)- 1 2 =1+a-f(a)- 1 2 & = 1 2 +a-f(a) & g(f(a))=-g(a) aligned Between any point a and f(a) there lies a root of g(x) ...(1) aligned & g(f(x))=-g(x) & g(f(x))+g(x)=0 & g(f(f(x))+g(f(x))=0 & g(f(f(x))=g(x) & g(1+x)=g(x) aligned g is periodic function with period 1....(2) Considering any n , n +1 interval, we can see from (1) and (2) that g(x) is identically zero aligned & g(x)=0 & f(x)=x+ 1 2 aligned

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