Find the domain of each rational expression. (3x + 7) / (4x + ...

Question

Find the domain of each rational expression. (3x + 7) / (4x + 2)(x - 12)

Accepted Answer

Hey everyone welcome back in this problem. For the following rational expression. Were asked to solve for the domain and were given nine x minus 37 divided by x minus 24 times X three X plus nine story. Okay, so we're gonna call this function F of X so that we can refer to it and it is equal to nine X minus 37 divided by x minus 24 times X three X three x plus nine. All right. So we want to find the domain of this function which means we want to find all of the X values where this function is defined where this function exists. Okay, Now, in order for the function to be defined, it needs to be the numerator of the function needs to be defined and the denominator of the function needs to be defined. Okay, there are more conditions but we need at least that. Okay, so let's take a look first at the domain of the numerator and the denominator individually. So in the numerator we have nine X -37. Okay. The domain of this Nine X minus 37. Well this is a linear function. Okay, it's a polynomial of degree one. We know with polynomial that they're defined everywhere. So their domain is from negative infinity to positive infinity. Okay, We can choose any X value. We want we can plug it into this function and it's going to give us a corresponding y value. Now in the denominator we have x minus 24 times three X plus nine. We look at the domain of that on its own. The domain of x minus 24 times three X plus nine. Okay, well this is a quadratic. Okay, it's also a polynomial is a polynomial of degree two. And again we know that polynomial are defined everywhere. They have a domain from negative infinity to positive infinity. Okay. We can choose any X value, plug it into this quadratic function and we will get a corresponding y value back. Okay, So the domain of the numerator and denominator is all the real numbers. Okay, so each of those functions on their own is defined everywhere. So that's great. Now we also need to look at where else could we have issues? Okay, Where could we have problems with a rational expression? And where we have problems is because of our division. Okay. We know that we can't divide by zero. So if the denominator is equal to zero then our function F is going to be undefined. Okay, So after backs is undefined When X -24 times three X plus nine is equal to zero. Okay, so we want to find the X values where this happens because they're going to be excluded from our domain. So this is already factored for us. The first factor X -24. That's going to equal zero. When X is equal to 24. Okay. And the second factor three X plus nine. Well this is going to equal to zero. We're going to get three X plus nine equals to zero. So three X is equal to negative nine and we get that X is equal to negative three. So our function f is going to be undefined when X is 24 or when X is negative three. Which means we need to exclude those from our domain. So when we're writing our domain, okay, It's gonna be D F. The domain of F. We know the only two points that we're gonna have issues are at our -3 and 24. Okay. So we can have from negative infinity all the way up to negative three. Ok. We're going to use a round bracket on negative three because we don't want to include the point negative three. Then we're gonna join this with the next interval. Okay. And the next interval is gonna start at negative three but not including negative three. So another round practice, we can choose any point all around negative three. We just can't be exactly at negative three. Ok. Is going to go up to 20 for our next problem point and we're gonna do the same thing. We're gonna join this with the interval from 24 onwards. Okay. So what we've done is we've chosen three intervals. Okay. And all of those is break up the function exactly at X equals 24 X equals negative three. Those points where the function is not defined. Okay, So our domain for a rational expression is going to be the union of the intervals from negative infinity to negative three, with negative three to negative 24 24 to infinity. All of those intervals do not include the endpoints. Okay. Alright. So if we look at our answer choices, we see that this is going to correspond with answer D. Thanks everyone for watching. I hope this video helped see you in the next one.

Find the domain of each rational expression. (3x + 7) / (4x + 2)(x - 12)

Key Concepts

Domain of a Rational Expression

Factoring and Setting Denominator Equal to Zero

Simplifying Rational Expressions

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