Find the domain of each rational expression. 12/ (x²

Question

Find the domain of each rational expression. 12/ (x² + 5x + 6)

Accepted Answer

Hey everyone in this problem for the following rational expression were asked to solve for the domain were given 23 divided by X squared plus 10 X plus 21. Okay so we're gonna call this function F of X just so that it's easier to refer to it. And it's gonna be 23 divided by X squared plus 10 X plus 21. Alright. We have this rational expression. We want to solve for the domain so we want to find all of the X values where this function is defined. Okay, now in order for the function to be defined we need the function in the numerator to be defined and we need the function in the denominator to be defined. Okay, so let's first look at the domain of each of those individually. So we look at the domain of the numerator. Yeah, this is gonna be the domain of 23. Well, 23 is just a constant. Okay, so it doesn't matter which X value. You choose the Y value is gonna be 23. It's going to be defined. It's going to exist. Okay, so the domain here is just gonna be from negative infinity to positive infinity. Any number in there. Now, if we look at the domain of the denominator. Okay. The domain of X squared plus 10 X plus 21 on its own. Okay, this is a polynomial function is a polynomial of degree two. So it's like a quadratic. We know that the domain of a polynomial is also all real numbers. So this domain is also going to be from negative infinity to infinity. Ok. You can choose any X value. You can plug it into X squared plus 10 X plus 21. This quadratic and you're going to get a corresponding Y value. Alright, so we have no issues with either the numerator or denominator. Not existing anywhere. Now for the domain of our function F we have to also consider is there going to be any issues somewhere? Okay, is there somewhere else in this function where something could go wrong where this function could become undefined? And when we have a rational expression we can get issues. If the denominator is equal to zero. Okay, if the denominator is equal to zero then you're going to be dividing by zero, which we know is going to make our function undefined. Okay, so our function F is going to be undefined when X squared plus 10, X plus 21 is equal to zero. Okay, so we want to find the X values when this happens. So let's factor our quadratic. Okay, this is just like finding the roots. We have a quadratic equal to zero. We want to find the roots. So if we factor we get X plus seven Times X-plus three is equal to zero. Okay, it's always a good idea to try to factor first. Um It's typically quicker. So if you see those factors factor that first. If you're getting stuck with the factors. Remember that? You can always use the quadratic formula to find the roots as well. Alright, so we have our factors X plus seven times X plus three. Okay, so X plus seven is going to be equal to zero when X is equal to negative seven and X plus three is equal to zero when X is equal to negative three. Okay, So we have that our function F is going to be undefined if X is negative seven or X is negative three. So when we write out our domain okay, those are the only points we have issues. So our domain is gonna be from negative infinity. All the way up to our first point. That's an issue. Which is negative seven. So we don't wanna include -7. We have a round bracket And then we're gonna take the interval from negative and seven up to negative three. Ok. We're just not including negative seven. Everything around negative seven is fine but right at negative seven. We're gonna have a problem. So, round brackets there we're going to do the same for the next interval Okay? With negative three and then all the way up to infinity. So we don't want to include negative three. So we use round brackets but everything right around negative three is fine. Okay. And so the domain of our function F is gonna be the union of the intervals negative infinity to negative seven. Negative 72 negative three and negative three to positive infinity. And if we look at our answer choices, we see that that's going to correspond with choice. See thanks everyone for watching. I hope this video helped see you in the next one.

Find the domain of each rational expression. 12/ (x² + 5x + 6)

Key Concepts

Rational Expressions

Domain of a Function

Factoring Quadratic Expressions

Watch next

Find the domain of each rational expression. 12/ (x2 + 5x + 6)

Key Concepts

Rational Expressions

Domain of a Function

Factoring Quadratic Expressions

Watch next

Find the domain of each rational expression. 12/ (x² + 5x + 6)