What is the domain and range of #y=3/(x-2)#?

1 Answer
Jul 24, 2017

Answer:

Domain: #(-oo, 2) uu (2, oo) " or " x != 2#
Range: #(-oo, 0) uu (0, oo) " or " y != 0#

Explanation:

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#.