Find the Domain of \(f(x)=\displaystyle\sum_{n=1}^{10}\frac1{|2x-n|}\)
Question
Find the domain of the function
\[ f(x)=\sum_{n=1}^{10}\frac1{|2x-n|} \]Solution
Given
\[ f(x)=\sum_{n=1}^{10}\frac1{|2x-n|} \]Since the denominator cannot be zero, we must have
\[ |2x-n|\ne0 \]for every value of \(n=1,2,3,\ldots,10\).
Therefore,
\[ 2x-n\ne0 \] \[ 2x\ne n \] \[ x\ne\frac n2 \]for
\[ n=1,2,3,\ldots,10 \]Thus the excluded values are
\[ \frac12,1,\frac32,2,\frac52,3,\frac72,4,\frac92,5 \]Hence the domain is all real numbers except these values.