How do you find the domain of f(x) = (3x+4) / (2x-3)f(x)=3x+42x3?

1 Answer
May 26, 2017

DOMAIN: {x|x!=3/2}{xx32}
RANGE: {y|y!=3/2}{yy32}

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=02x3=0
2x=32x=3
x=3/2x=32

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

Domain: All real numbers except x=3//2x=3/2. More formally, you could state the domain as {x|x!=3/2}{xx32}.

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=0y=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=y= that fraction. So in our case, you have a horizontal asymptote at y=3/2y=32