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

May 26, 2017

DOMAIN: $\left\{x | x \ne \frac{3}{2}\right\}$
RANGE: $\left\{y | y \ne \frac{3}{2}\right\}$

#### 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

$2 x - 3 = 0$
$2 x = 3$
$x = \frac{3}{2}$

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 $\left\{x | x \ne \frac{3}{2}\right\}$.

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 $y = \frac{3}{2}$