In general, given a function g(x), a vertical shift is given by the transformation below
Therefore, f(x) is the graph of 2/(x-3) translated 1 unit upwards.
Notice that 2/(x-3) is a rational function; then, we need to find its asymptotes as shown below
As for the horizontal asymptote,
However, we need to translate the graph upwards by one unit; then, the horizontal asymptote becomes y=1.
Knowing the asymptotes, we can graph the function, as shown below
In more detail
Finally, as for the domain and range of the function, the asymptotic values are not included; therefore,