Find the Domain of \(f(x)=\dfrac1{\sqrt{[x]^2-3[x]+2}}\)
Question
Find the domain of the function
\[ f(x)=\frac1{\sqrt{[x]^2-3[x]+2}} \]where \([x]\) denotes the greatest integer function.
Solution
Given
\[ f(x)=\frac1{\sqrt{[x]^2-3[x]+2}} \]Since the square root is in the denominator, we require
\[ [x]^2-3[x]+2>0 \]Let
\[ [x]=t \]Then,
\[ t^2-3t+2>0 \]Factorize:
\[ (t-1)(t-2)>0 \]Therefore,
\[ t<1 \quad \text{or} \quad t>2 \]Since \(t=[x]\),
Case 1: \([x]<1\)
\[ x<1 \]Case 2: \([x]>2\)
Since \([x]\) is an integer,
\[ [x]\ge3 \]Therefore,
\[ x\ge3 \]Combining both cases,
\[ x<1 \quad \text{or} \quad x\ge3 \]