Given f(x)=sin−1[2x2−3]+log2(log21(x2−5x+5))
For f(x) to be defined
−1≤[2x2−3]<1
⇒−1≤2x2−3<2
⇒2<2x2<5
⇒1<x2<25
i.e. x∈(−25,−1)∪(1,25)...(i)
and x2−5x+5>0
⇒(x−(25−5))(x−(25+5))>0
i.e. (−∞,25−5)∪(25+5,∞)...(ii)
Also log21(x2−5x+5)>0
⇒x2−5x+5<1
⇒x2−5x+4<0
i.e. x∈(1,4)...(iii)
From (i),(ii)&(iii)
So, x∈(1,25−5)