SOLUTION: My teacher has told me that it is impossible to find a domain for equations like: sqrt(-1-2x) sqrt(-2x-4) I can't understand why the domain would not be (-∞, -½) and (

Algebra ->  Radicals -> SOLUTION: My teacher has told me that it is impossible to find a domain for equations like: sqrt(-1-2x) sqrt(-2x-4) I can't understand why the domain would not be (-∞, -½) and (      Log On


   

Question 443978: My teacher has told me that it is impossible to find a domain for equations like:
sqrt(-1-2x)
sqrt(-2x-4)
I can't understand why the domain would not be (-∞, -½) and (-∞, -2). I was told that finding a domain was impossible because both elements in the equation were negative (-1 and -2x; -2x and -4). However, because x is an unknown and could be negative, I don't quite see how what I was told is possible. Is the variable x not considered when finding domains?
Answer by swincher4391(1107) About Me  (Show Source):
You can put this solution on YOUR website!
It doesn't sound like your teach knows what they're talking about.
There are multiple ways to find the domains of these functions:
One find the inverse of the function's range (which is the original's domain)
Two solve for where x is defined.
===============
For sqrt%28-1-2x%29 it is defined strictly where -1-2x+%3E=+0
So solve for x to get -1+%3E=+2x -1%2F2%3E=x%7D%7D+%7B%7B%7Bx%3C=+-1%2F2
From -infinity to -1%2F2 is the domain.
===============
You can find the inverse as well. y+=+sqrt%28-1-2x%29 y%5E2+=+-1-2x
y%5E2+%2B+1+=+-2x -1%2A%28y%5E2%2B1%29%2F2+=+x
Switch the xs and ys to get
y+=+expr%28-1%2F2%29%28x%5E2%2B1%29
Since the range of %28x%5E2%2B1%29 is x%3E=1 then if you multiplied all values by -1/2, you'd get x%3C=-1%2F2 which is the same as what we had.
===================
I believe the argument you teacher may have tried to make is:
sqrt%28-1-2x%29+=+sqrt%28-1%2A%281%2B2x%29%29+=+sqrt%28-1%29+%2A+sqrt%281%2B2x%29
Since sqrt%28-1%29 is not allowed, then you can't find the domain. But this is not true. sqrt%28-1%2A%281%2B2x%29%29 is defined where %281%2B2x%29%3C=+0, because a negative times a negative gives us a positive (which we can take the square root of). So %281%2B2x%29+%3C=+0 +1+%3C+-2x -1%2F2+%3E=+x x+%3C=+-1%2F2... again gives us the same result.
================
Thus you can find the domain.
===============
Does this make sense? I hope this helped.