what is the domain of g(x) = square root of (x2 + 2x - 8)

Question

please help!!

Algerbra 2

Answer 1

We know that we can't take the square root of a negative number.  So you know your expression must be positive

 

• x²+2x-8 > 0
• (x+4)(x-2) > 0  ~ First step is to factor
• Note that this expression EQUALS 0 for x=-4 and x=2.  These are important numbers
• Note that x=-4 and x=2 break the number line into three distinct regions.  We now need to test all three regions to see if the results are positive or negative
• First region, x<-4 
• Pick any point, I'll use -5.  Using the factored form (because it's easier) this gives (-1)*(-7), which is positive (note: we really only care about the sign)
• Second region -4<x<2
• Always pick 0 if it's an option, as it's easy.  This gives (4)*(-2), which is negative
• Third region x>2
• Choose 3 to get 7*1, which is positive
• Back to the original equation, we WANT positive responses, so the first and third regions are part of the domain, but not the second.  We can include 0
• Final answer: Domain = (-infinity, -4] U [2, infinity)
• Or written a different way, x < -4 OR x > 2