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) = log[(x+1)/(x-5)]

Accepted Answer

find the domain of the following function F. Of X equals log x minus three over X plus seven. Now the final domain of this we want to determine where the part inside our log is greater than zero. We'll call this G. X. So we have flog G. X. Where G. X. Is equal to X months three over X plus seven. Now our domain is going to be given when G. X. Is square than zero. This is because we can't have a negative in our luck. So we're gonna find where G. X. Is greater than zero. So let's take X -3 Over X-plus seven. Find where it's greater than syrup concerning this. Gonna say expands three X plus seven equals zero. And we're gonna set our numerator equal to zero and the denominator equal to zero. So let's say X -3 equals zero. We get x equals three. X plus seven equals zero. We get X equals negative seven. So we have two points where We're gonna be equal to zero at and -7. Or rather we have one point of schools here and one is actually undefined at X equals negative seven. So we're actually gonna check our points here. So I'm gonna make the number line to determine what are signs are of our graph. Now we have three and -7. Now I want to determine N G. M X. Where are we greater than zero. Let's say we have X equals zero here. X equals five and X equals negative 10. We'll play the sensor G X. Well so you want G of negative 10. That's equal to negative 10 minus three over negative 10 plus seven. That is negative 13 over negative three. Just 13/3. That means that we have a positive let's say one G zero G zero. That means we have zero minus 3/0 plus seven which is negative 3/7. That is a negative 31 G five. That is 5 -3 over five plus seven. First gives us to over 12. Just 1/6. That is positive. So we notice here we have positive until we get to negative seven. Then we go negative for a little bit. Once we hit three we go after positive. We can write our interval now our interval we can assume from native infinity until we get to negative seven we will be greater than zero or positive. And now we're negative until we've got two negative three. I'm sorry, positive three. So from positive three to infinity we are positive again. This would be our domain which means the answer to our problem. His answers. 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) = log[(x+1)/(x-5)]