Answer
Domain = $\mathbb{R}\backslash\{1\}$ or $\{x\in \mathbb{R}\ |\ x\neq 1\}$
Work Step by Step
The domain of a real function ( a function defined on real numbers)
is the set of numbers for which f(x) is defined.
$ f(x)=\displaystyle \frac{x}{x^{2}-2x+1}\quad$ can be calculated for any real number,
except for those yielding a zero in the denominator.
Here,
$ x^{2}-2x+1=0\quad$ ... recognize a perfect square
$(x-1)^{2}=0$
$x=1 \quad $must be excluded from the domain.
Domain = $\mathbb{R}\backslash\{1\}$ or $\{x\in \mathbb{R}\ |\ x\neq 1\}$