f(x)=root(log_0.3(x-1)/x²-2x-8)

The domain of f(x) is :

necessary condition : (x-1)>0

the value under f(x) shld always be positive

case 1:

log0.3(x-1) > 0  and (x+2)(x-4) > 0

x-1 > 1  and (x+2)(x-4) > 0  and x>1

the overlapping region is x > 4

case 1:

log0.3(x-1) < 0  and (x+2)(x-4) < 0

x-1 < 1  and (x+2)(x-4) < 0

x<2 and -2<x<4  and x>1

the overlapping region is : (1 , 2)

the domain is : (1,2) U (4,infinity)

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Regards

Ramesh