Given: #3/(x-2)#
The domain is the valid input, #x#.
The given equation is a rational function #y = (N(x))/(D(x)) = (a_nx^n + ...)/(b_mx^m + ...) #
If #D(x) = 0# the function will be undefined.
Where the function is undefined, you will have a vertical asymptote.
If we set #D(x) = 0# we will find where the function is undefined:
#x - 2 = 0; " so " x = 2 # is where the function is undefined. This means the domain cannot include #x = 2#.
The range is based on the degree of the polynomials:
When #n < m" we have a horizontal asymptote at "y = 0#
When #n = m " we have a horizontal asymptote at "y = a_n/b_m#
When #n > m# there is no horizontal asymptote.
In the example, #n = 0 " and " m = 1 " so " n < m#: we have a horizontal asymptote at #y = 0#
This means #y != 0#.
