Question

How do you find the domain and range of f(x)= 8/x-4 in Interval Notation?

Answer 1

Domain: `(-oo, 0) uu (0, oo)`

Range: `(-oo, -4) uu ( -4, oo)`

Explanation:

Given: `f(x) = 8/x - 4`

Analytically you would first find a common denominator:

`f(x) = 8/x - 4/1 * x/x = (8-4x)/x = (-4x + 8)/x = (-4(x - 2))/x`

This type of equation is called a rational (fraction) function: `(N(x))/(D(x))`

When `D(x) = 0` we can find vertical asymptotes. They limit the domain of the function.

There is a vertical asymptote at `x = 0`. This means that `x=0` is not included in the domain.

A domain of all reals in interval notation is `(-oo, oo)`

We use the set notation of union `uu` to tie the two pieces together:

Domain: `(-oo, 0) uu (0, oo)`

Horizontal asymptotes limit the range of the function. To find horizontal asymptotes we look at the degree of both the numerator and denominator: `(N(x))/(D(x)) = (a_nx^n + ....)/(b_nx^m+...)`

If `n < m`, the horizontal asymptote is `y = 0`

If `n = m`, the horizontal asymptote is `y = a_n/b_m`

If `n > m` there is no horizontal asymptote

In the given function `n = m = 1`

The horizontal asymptote: `y = -4/1; " "y = -4`

Range: `(-oo, -4) uu ( -4, oo)`

graph{8/x - 4 [-10,10, -15, 15]}