Find the domain of each rational expression. See Example 1. (3...

Question

Find the domain of each rational expression. See Example 1. (3x + 7) / (4x + 2) (x - 1)

Accepted Answer

Hello, everyone. We are asked for the following rational expression to determine its domain. The expression we are given is the parentheses 28 X plus 13 divided by open square bracket. The parentheses nine X plus three multiplied by the parentheses, X minus three closed square bracket. Our answer choices are a, the interval from negative infinity to negative one third, closed parentheses. Union with the interval, open parentheses three to positive infinity. B the interval from negative infinity to negative one third parentheses. Union with the parentheses negative one third to negative 13, divided by 28 closed parentheses. Union with the interval, open parentheses, negative 13 divided by 28 to 3 closed parentheses. Union with the interval, open parentheses, three to positive infinity. C the interval negative infinity to negative one third, closed parentheses. Union with the interval from open parentheses, negative one third to three closed parentheses. Union with open parentheses three to positive infinity. D the interval from negative infinity to negative 13, divided by 28 closed parentheses. Union with open parentheses negative 13 divided by to positive infinity. So let's rewrite this as a fraction. I think that might help us see a little bit more what we're working with and we can cut down on some of our grouping symbols. So 28 X plus divided by, and my denominator is nine X plus three in parentheses, multiplied by X minus three in parentheses. So I recall my domain is all the X values except whatever X values will make my denominator equal zero. So it'll be undefined when our denominator equals zero. So to find out where our domain is undefined, we are gonna set our denominator equal to zero. So nine X plus three in parentheses, multiplied by the parentheses X minus three equals zero. So we can find where this is undefined. Recall that when we have factors that multiply to zero, we can solve this by setting each factor individually equal to zero. So nine X plus three would equal zero and X minus three would equal zero. So solving nine X plus three equals zero, I subtract three from both sides. We have nine X equals negative three, dividing both sides by nine, we get X equals negative one third. So this will be undefined when X equals negative one third. So it cannot equal that. And then for X minus three equals zero, I add three to both sides. So this would be undefined when X equals three. So we don't want X to equal three. So we now know where the gaps in our domain are. So let's write this out as intervals. So we open a parentheses and we start at negative infinity and it'll go all the way until we hit, hit negative one third, which is not included. So we're gonna close parentheses and then it's gonna be a union and then we'll have the interval that runs between negative one third and three. But both of those are parentheses because those end points are not included. And we're union that with the interval that goes from positive three to positive infinity. And again, we have parentheses. So we have skipped over the parts that don't work that give us an undefined answer. And those were the values negative one third and three. And so we get an interval from negative infinity to negative one third and then negative one third to three and then three to positive infinity without negative one third or three being included. And that matches answer choice C have a nice day.

Find the domain of each rational expression. See Example 1. (3x + 7) / (4x + 2) (x - 1)

Key Concepts

Domain of a Rational Expression

Factoring and Setting Denominator Equal to Zero

Simplifying Rational Expressions

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