Question

What is the domain and range of `f(x)= (x+1)/(x^2+3x-4)`?

Answer 1

Domain: `RR-{4, +1}`
Range: `RR`

Explanation:

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

Notice that the denominator can be factored as
`color(white)("XXX")(x+4)(x-1)`
which implies that the denominator would be `0` if `x=-4` or `x=1`
and since division by `0` is undefined
the Domain must exclude these values.

For the Range:
Consider the graph of `f(x)`
graph{(x+1)/(x^2+3x-4) [-10, 10, -5, 5]}
It seems clear that all values of `f(x)` (even within `x in(-4,+1)`) can be generated by this relation.
Therefore the Range of `f(x)` is all Real numbers, `RR`