Find the domain of each rational expression. 3 / x2 - 5x - 6

Question

Accepted Answer

Hey everyone for the following rational expression were asked to solve for the domain were given 19 divided by x squared minus 12 X minus 13. Alright, so let's go ahead. We're just going to call this f of X. OK, so that we can refer to it and this is going to be 19 divided by x squared minus 12 x minus 13. Now when we're looking at a rational expression we have a numerator denominator. Okay. And we're looking for the domain, we wanna find everywhere where this function exists. So let's start with the numerator. OK. The numerator is just 19. Okay, so the domain of the numerator. So the domain of 19. This is just going to be all real numbers or anything if we're looking at X. Okay, it's just a constant. So it doesn't depend on the value of X. You can put any X value, choose any X value you want in this function is to find if we're just looking at the numerator, 19. Okay, let's do the same for the denominator. The domain of x squared minus 12 X minus 13. Okay, well this is a polynomial. It's a polynomial degree two. We know that polynomial are also defined for all X values. Okay, so this also is going to have a domain from negative infinity to positive infinity. You can choose any X value. You can put it into this polynomial function. This quadratic function and you're gonna get a corresponding Y value. Alright, so we know that the domain of the numerator and denominator is everything all real numbers from negative infinity to positive infinity. Okay. So we know that there's gonna be no issues there with numbers not existing or values not existing for either of those. Okay. What we want to look at though is we're dividing. Okay. So we want to see where where else could we have problems? Where else could this function be undefined? Okay. And what we'll see is that this function F is going to be undefined? Okay. If the denominator is equal to zero, okay. If the denominator is equal to zero, we're dividing by zero. Okay. And we know that the function is gonna be undefined if we're dividing by zero. So F is going to be undefined when X squared -12, X -13 is equal to zero. Now, trying to figure out when this is equal to zero, This is like a factoring finding roots problem. Okay, so we can factor X squared minus 12 X minus as x minus 13. Okay. Times X plus one, this is gonna equal to zero. And if you're stuck there, if you can't find those factors, okay, you can always use the quadratic formula to solve for those roots as well. Okay, sometimes you look at a quadratic and the factors just aren't working out for you, whatever the case is, you can always use the quadratic formula. It's going to take a little bit longer. But if you're stuck it's always an option. Alright. So we get our roots the first factor X -13. That's going to be zero when X is 13 and the second root X plus one. That's going to be +01. X is equal to negative one. Okay, so our function is going to be undefined when X is 13 or X is negative one. So to write out our domain we're going to say D F. The domain of F. We're going to have the phone, we know that we can go from negative infinity All the way up to -1. Okay. We know there are no issues there because our numerator denominator exists. The denominator does not equal zero. Now we're going to take the union of this with the next interval. Okay, what is the next interval? Well we're actually going to start right at negative one again as long as we don't include exactly negative one. Okay, we're okay. So we use round brackets to indicate that we don't include negative one And our next interval is going to be negative one up to that next point where we have an issue which is And then we do the same thing on the other side. Okay, from 13 onwards up to infinity this function is defined again. Alright, so the domain of our function is going to be the interval negative infinity negative one. The union with the interval from negative 1 to 13 and the union with the interval from 13 to infinity. All of those intervals are not inclusive of the endpoints. So we look at our answer choices. We see that we have answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.

Find the domain of each rational expression. 3 / x2 - 5x - 6

Key Concepts

Rational Expressions

Domain of a Function

Factoring Polynomials

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