Question

find the range of the following functions

g(x) =x/x-2

Answer 1

If you mean
g(x) = x/(x-2) then the range is all real values of y except y = 1

Answer 2

how did you get that and how would you graph this function

Answer 3

as x --> ±∞
the value of x/(x-2) approaches 1, but no value of x will give us 1
try solving 1 = x/(x-2)
x-2 = x
0 = 2 , which is a contradiction, so no value of x makes x.(x-2) equal to 1

e.g. let x = 1000000
then g(x) or y = 1000000/999998 = 1.000002
let x = -1000000
then y = -1000000/-1000002 = .999998

so y = 1 is a horizontal asymptote
there is also a vertical asymptote at x=2, since x=2 makes y undefined.

the graph will be a hyperbola with major axis as y = x - 1 and centre (2,1)