Rewrite the numerator as
<em>x</em> ² + 3<em>x</em> + 5 = (<em>x</em> - 1/2)² + 4 (<em>x</em> - 1/2) + 27/4
Then
(<em>x</em> ² + 3<em>x</em> + 5) / (2<em>x</em> - 1) = 1/2 × (<em>x</em> ² + 3<em>x</em> + 5) / (<em>x</em> - 1/2)
… = 1/2 × ((<em>x</em> - 1/2)² + 4 (<em>x</em> - 1/2) + 27/4) / (<em>x</em> - 1/2)
… = 1/2 × ((<em>x</em> - 1/2) + 4 + 27 / (4 (<em>x</em> - 1/2)))
… = 1/2 <em>x</em> + 7/4 + 27 / (8 (<em>x</em> - 1/2))
which clearly has a non-removable singularity at <em>x</em> = 1/2, which is to say this function has a domain including including all real numbers except 1/2.
For every number other than <em>x</em> = 1/2, the function takes on every possible real numbers, since 1/2 <em>x</em> + 7/4 alone takes on all real numbers.
So:
domain = {<em>x</em> ∈ ℝ | <em>x</em> ≠ 1/2}
range = {<em>x</em> ∈ ℝ}
