range for g(x)?

Question

range for g(x) = (x-3) / (x-1) for x cannot be 1..
range is R \ {1}...is this correct?..if so then why?

Answer 1

Let (x-3) / (x-1) = y, where x ≠ 1.
Then x - 3 = y(x - 1)
so (y-1)x = y - 3
so x = (y-3) / (y-1) if y ≠ 1.
(Note that in this case, g(x) is its own inverse. This is not normally true.)
So for any y ≠ 1 there is an x such that g(x) = y. So the range of g(x) contains R \ {1}.
If g(x) = 1 then x-3 = x-1; this is impossible. So there is no x such that g(x) = 1. So the range of g(x) does not contain 1.
Hence the range of g(x) is R \ {1}.

Answer 2

No. x ε [R \ {1}] is the domain of x. The range of g is all of the values that g may assume as x runs over its domain. In this case, range(g) = R.

HTH

Doug