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

Question

Accepted Answer

Hey everyone welcome back for the following rational expression were asked to solve for the domain were given nine X minus 37 divided by x plus 37. So let's just go ahead and call this the function F of X just so that we can refer to it with the name. And this is gonna be nine x minus 37 divided by X plus 37. Alright, so when we have a rational expression, one thing divided by another thing. But we want to do, when we're looking for the domain we want to find all of the X values where this function exists where it's defined. Okay, in order for that to happen, we need the function in the numerator to be defiant and we need the function in the denominator to be defined. Okay, so the first thing we want to do is look at the domain of the numerator and the domain of the denominator individually. So if we look at the domain the numerator, which is the domain of nine X minus 37. Okay, this is a linear equation. Okay. It's a polynomial of degree one. And we know with polynomial that the domain is gonna be from negative infinity to infinity. Ok. This is defined for all X values. You can choose any X value, plug it into this linear equation and you're gonna get a corresponding y value. We're gonna do the same for the denominator and the domain of X plus 37. Again, this is a linear equation polynomial of degree one. So same thing, it's defined for all X values negative infinity to positive infinity. You can choose any X value, plug it into that equation and you'll get a corresponding Y value. Alright, so we're not gonna have any issues where the domain or the denominator or sorry, the domain of the numerator or the domain of the denominator don't exist where those functions don't exist. So we don't have any issues like that. Where else could we get issues with rational expressions? Okay, now we're dividing by something and we know that if we get division by zero then our function will be undefined. Okay, so f of X is going to be undefined when the denominator is equal to zero. So when X plus 37 is equal to zero. So we want to figure out what X. Value this happens at so that we can exclude it from our domain. Well this is going to happen when X is equal to -37. So when X is -37, our function is undefined. So if we want to write out our domain it's gonna be D F. The domain of F. Okay, well we know that this is gonna be from negative infinity and we can go all the way up to negative 37 but we can't include negative 30 seconds. We're gonna use a round bracket to indicate that negative 37 is not included. And then we're gonna take the union of the interval from negative up to infinity. Ok so we've just broken up the interval from negative infinity to positive infinity at negative 37. And we've used round brackets on each end to indicate that we don't include negative 37. Okay. We can have X values all around negative 37. Right up until negative 37 but not exactly at negative 37. Okay. Otherwise we're dividing by zero in our function is not going to exist. So our domain is going to be the union of the two intervals from negative infinity to negative 37 from negative 37 to positive infinity where all the endpoints are not included. Okay, if we look at our answer choices, we see that this is going to correspond with answer. A thanks everyone for watching. I hope this video helped see you in the next one.

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

Key Concepts

Rational Expressions

Domain of a Function

Finding Restrictions

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