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

1 Answer
May 25, 2016

Answer:

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}.