How do you find the domain of #f(x) = (3x+4) / (2x-3)#?

1 Answer
May 26, 2017

Answer:

DOMAIN: #{x|x!=3/2}#
RANGE: #{y|y!=3/2}#

Explanation:

The domain consists of all numbers you can legally plug into the original. The excluded "illegal" values would be dividing by zero or negatives under square roots.

This expression has a denominator, so there is a risk of illegally dividing by zero. This would happen only if

#2x-3=0#
#2x=3#
#x=3/2#

This means that #x=3//2# is excluded from the domain. Therefore,

Domain: All real numbers except #x=3//2#. More formally, you could state the domain as #{x|x!=3/2}#.

For rational functions, you find the range by evaluating the degree of the numerator compared to the degree of the denominator. If the degree of the top > degree of bottoms, then you have a horizontal asymptote at #y=0#. If they are equal, you have a horizontal asymptote. The coefficient of highest degree in the numerator is divided by the coefficient of the highest degree on the bottom. The result is #y=# that fraction. So in our case, you have a horizontal asymptote at #y=3/2#