Question

What is the domain and range of `(x+5)/(x+1)`?

Answer 1

Domain = `RR-{-1}`
Range = `RR-{1}`

Explanation:

First of all, we must note that this is a reciprocal funtion, as it has `x` in the lower part of the division. Therefore, it will have a domain restiction:

`x+1 !=0`
`x!=0`

The division by zero is not defined in mathematics, so this function will not hava a value associated to `x=-1`. There will be two curves that pass near this point, so we can procced to plot this function for points around this restriction:

`f(-4)=1/-3=-0.333`
`f(-3)=2/-2=-1`
`f(-2)=3/-1=-3`
`f(-1)=cancel(EE)`
`f(0)=5/1=5`
`f(1)=6/2=3`
`f(2)=7/3=2.333`

graph{(x+5)/(x+1) [-10, 10, -5, 5]}

There is also a hidden range restriction in this function. Notice that the curves will keep going towards the infinitity in both sides by the x axis, but they never reach a value. We must calculate the limits of the function in both infinities:

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

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

This number can be found if you solve the function for a very big number in x (1 million, for example) and a very small number (-1 million). The funcion will get near `y=1`, but the result will never be exactly 1.

Finally, the domain can be any number, except -1, so we write it this way: `RR-{-1`.
The range can be any number except 1: #RR-{1}.