Method 1
If you graph the function, it would look like:
graph{1/(x-3) [-10, 10, -5, 5]}
By looking at the graph, you can see that #x# can be any number. However, even though the #x# values get closer and closer to #3#, it never reaches #3#. Thus, the domain is #color(green)(|bar(ul(color(white)(a/a){x in RR|x!=3}color(white)(a/a)|)))#.
Similarly, you can see that #y# can be any number as well. However, as the #y# values approach #0#, they only get closer, but never actually reach #0#. Thus, the range is #color(green)(|bar(ul(color(white)(a/a){y in RR|y!=0}color(white)(a/a)|)))#.
Method 2
To determine the domain, set the denominator of the function to cannot equal #0# and solve for #x#. The result is the restriction part of the domain.
#x-3!=0#
#x!=3#
Thus, the domain is #color(blue)(|bar(ul(color(white)(a/a){x in RR|x!=3}color(white)(a/a)|)))#.
Recall that #y=1/(x-3)# can be written as #y=1/(x-3)+0#. The #+0# indicates the restriction for the #y# values of the function. It states that #y!=0#.
Thus, the range is #color(blue)(|bar(ul(color(white)(a/a){yin RR|y!=0}color(white)(a/a)|)))#.
