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

Jul 24, 2017

Domain: $\left(- \infty , 2\right) \cup \left(2 , \infty\right) \text{ or } x \ne 2$
Range: $\left(- \infty , 0\right) \cup \left(0 , \infty\right) \text{ or } y \ne 0$

#### Explanation:

Given: $\frac{3}{x - 2}$

The domain is the valid input, $x$.

The given equation is a rational function $y = \frac{N \left(x\right)}{D \left(x\right)} = \frac{{a}_{n} {x}^{n} + \ldots}{{b}_{m} {x}^{m} + \ldots}$

If $D \left(x\right) = 0$ the function will be undefined.

Where the function is undefined, you will have a vertical asymptote.

If we set $D \left(x\right) = 0$ 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 $n < m \text{ we have a horizontal asymptote at } y = 0$

When $n = m \text{ we have a horizontal asymptote at } y = {a}_{n} / {b}_{m}$

When $n > m$ there is no horizontal asymptote.

In the example, $n = 0 \text{ and " m = 1 " so } n < m$: we have a horizontal asymptote at $y = 0$

This means $y \ne 0$.