Question

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

Answer 1

Domain: `(-oo, 2) uu (2, oo) " or " x != 2`
Range: `(-oo, 0) uu (0, oo) " or " y != 0`

Explanation:

Given: `3/(x-2)`

The domain is the valid input, `x`.

The given equation is a rational function `y = (N(x))/(D(x)) = (a_nx^n + ...)/(b_mx^m + ...)`

If `D(x) = 0` the function will be undefined.

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

If we set `D(x) = 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" we have a horizontal asymptote at "y = 0`

When `n = m " we have a horizontal asymptote at "y = a_n/b_m`

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

In the example, `n = 0 " and " m = 1 " so " n < m`: we have a horizontal asymptote at `y = 0`

This means `y != 0`.