Question

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

Answer 1

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`