What is the domain in interval notation of g(x)=log(x+3/x+4)

Question

Please answer my question step by step find the domain in interval notation

Answer 1

x ≠ -4, that would make the denominator zero, a definite no-no.

x ≠ -3, that would make the numerator zero. No power on a base can result in zero.

-4 < x < -3 would make the fraction negative. No power on a base can result in a negative.

 

(oo, -4) ∪ (-3, oo)

Answer 2

Two things Edgar, (x+4) can't be zero and the whole rational function can not be negative or zero.

(x+3)/(x+4)>0?

 

x+3>0 when x>-3

x+4>0 when x>-4

 

 
x+3<0 when x<-3
x+4<0 when x<-4

 

Notice how the top and the bottom have the same sign (+ and +) when x>-3, and when x<-4, and that the top (x+3) and the bottom (x+4) have opposite sign when -4<x<-3

Therefore sol. x<-4 or x>-3