In Exercises 109–112, find the domain of each logarithmic functio...

Question

In Exercises 109–112, find the domain of each logarithmic function. f(x) = ln (x² - x − 2)

Accepted Answer

find the domain of the following function F of X equals Ln expert minus two X minus eight. So let's go ahead and write this up. Now if Ellen I'm going to say G X where G X is our function inside the Ellen X squared minus two. X minus eight. To determine the domain. We want to find where Ellen is not negative or in other words we want to find where G X is not negative. We'll say greater than zero. L n g X. Let's figure out where our function G X is greater than zero. Some we want x squared minus two X minus eight greater than zero. So find that we're gonna go ahead and solve for X. Let's say X squared minus two, X minus eight equals zero. That's the software X. We're gonna factor you have a text here. X here I want to find factors add up to equal negative two and multiply two equal eight and negative eight. In this case we're going to have a positive and a negative side because we're going to negatively now eight We have one and 8, two and four. We want to add up to -2. So we'll say we use two and four. Let's see with you X plus two. X minus four. These are two factors. We want to determine where these equal zero. So we'll say X plus two equals zero and x minus four equals zero, solving for X. We get X equals negative two, X equals positive four. So we have where we're equal to zero but now I'm going to find out when we're positive. So I'm gonna make a number line to do this where we have X. Is for X. Is negative. To even to find numbers and those intervals to determine what the sign is. Let's say X zero here because zero is between the age of two and four, X is five because it's bigger than four. X is -3. Let's go ahead and solve G. G of negative three is first city of negative three negative three squared minus two times negative three minus eight. Simplify this nine plus six minus P. Which gives us a positive 70. So that is positive. Let's let X zero now G zero that is zero squared minus two times zero minus eight which is negative eight. That is negative. Finally G five that is five squared minus two times five minus eight. That would be 25 -10 -8. That is equal it's a positive step. So you're positive. So you want to find the interval where it's positive if you look here until we get to negative two were positive. So we have negative infinity It's a negative two and From four on more positive so it's a four to infinity you're positive. This is our interval and we can't simplify it anymore. The answer to our problem. It's answered Okay I hope to help you solve the problem. Thank you for watching. Goodbye

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In Exercises 109–112, find the domain of each logarithmic function. f(x) = ln (x² - x − 2)