Question

How do you find the domain and range of `(x^2-64)/(x-8)`?

Answer 1

Domain: `RR "\" { 8 } = (-oo, 8) uu (8, oo)`

Range: `RR "\" { 16 } = (-oo, 16) uu (16, oo)`

Explanation:

`f(x) = (x^2-64)/(x-8) = ((x-8)(x+8))/(x-8) = x+8`

with exclusion `x != 8`

When `x=8`, both the numerator and denominator are `0` resulting in an undefined value. So `x=8` is not in the domain.

`f(x)` is well defined for all other values of `x`

So the domain is `RR "\" { 8 } = (-oo, 8) uu (8, oo)`

The range is the set of values that `f(x)` takes for `x in RR "\" { 8 }`, namely `(-oo, 16) uu (16, oo)`.

Note that `f(x)` can take any value except `16`