Question

What are the domain restrictions of `(x^2 - 3x - 4)/(x^2 - 7x - 8)`?

Answer 1

`{x: x!= 8}`

Explanation:

any number divided by `0` is undefined.

here, the domain restrictions exclude the values of `x` for which the denominator would be `0`.

`-8 + 1 = -7`
`-8 * 1 = -8`

`x^2-7x-8 = (x-8)(x+1)`

`-4 + 1 = -3`
`-4 * 1 = -4`

`x^2-3x-4 = (x-3)(x+1)`

`(x^2-3x-4)/(x^2-7x-8) = ((x-3)(x+1))/((x-8)(x+1))`

`= (x-3)/(x-8)`

if `x-8 = 0`, then `x = 8`

for the `x`-value of `8`, `y` cannot be defined.

the domain can therefore be written as `{x: x!= 8}`

here is the graph:
graph{(x-3)/(x-8) [2.77, 12.77, -1.6, 3.4]}