Find the domain in interval notation

Question

What is the domain in interval notation of 

y=ln(x^2-1x-20)

 

Answer 1

In this equation, you have a natural log of some number. This is the same as log base e of that number. Since e is not 0, the only limitation on x is that (x^2-1x-20) cannot be less than or equal to zero, since e to any power will never equal 0 or be negative.

 

All we need to do here is set what is in the parentheses equal to 0 to find what x cannot equal, then determine where x would also be negative.

 

0=x^2-x-20  (1x is the same thing as x)

0=(x-5)(x+4)   (I unFOILed this by considering possible numbers that multiply to equal 20 [1,20; 2,10; 4,5] and then found the combination that would equal -1 when one was negative and one was positive)

 

From this, we see that if x=5 or x=-4, the equation will be true. (5-5)(5+4)=0*9=0  (5-(-4))(-4+4)=9*0=0

 

So we know that x CANNOT equal either 5 or -4. Now we need to test to see where it would be negative. Use the steps from the previous problem to test this. For simplicity, I'm just going to state that x would be negative between -4 and 5. So interval notation of the answer would look like this:

 

(-∞,-4) U (5,∞)