What is the domain of #g(x) = sqrt(-3x - 2)#?

1 Answer
Feb 4, 2016

Given function is #g(x)=\sqrt{-3x-2}#

Now, one thing for sure is that we can't have a real valued function involving imaginary outputs. So that means the values under the square root should be positive.

So, from the function itself #\sqrt{-3x-2}=0# is the least possible value. Squaring and rearranging on both sides we get
#-3x=2\impliesx=(-2)/3#

But what side of the real numbered line should be included? Left side of #-2/3# or the right?
So again, we take the square root, and then square and rearrange, but this time, we say that the function inside the square root should be positive.

In a more simple way of saying it mathematically
#-3x-2>0\implies-3x>2\impliesx<-2/3# (I've multiplied the inequality by a negative value, that's why the sign reverses).

So in the end, #x# belongs to the real number set, containing the values #(oo,(-2)/3]#