Question

How do you find the domain and range of `f(x) = (x-4)/(x+3)`?

Answer 1

The domain is `x!=-3` as this would make the denominator `=0`.
There are no other restrictions.

Explanation:

For the range we look at what happens when `x->+-oo`
In both cases the fraction will approach `f(x)=+1`, but will never get there. `y!=+1`

In short:
`lim_(x->-3^-)f(x)=+oo` and `lim_(x->-3^+)f(x)=-oo`

`lim_(x-> -oo)f(x)=lim_(x->+oo)f(x)=+1`

`x=-3` and `y=+1` are called the asymptotes .
graph{(x-4)/(x+3) [-25.65, 25.65, -12.82, 12.84]}