In Exercises 1–30, find the domain of each function. f(x) = (2x+7...

Question

In Exercises 1–30, find the domain of each function. f(x) = (2x+7)/(x^3 - 5x^2 - 4x+20)

Accepted Answer

Welcome back. I'm so glad you're here. We have this function F of X equals seven X plus four divided by x cubed plus x squared minus 49 X minus 49. And we are asked to find the domain. Well, what numbers can X be here? We have a restriction because there's a fraction in our function. And the restriction for fractions is that the denominator cannot equal zero. If the denominator equals zero, then the fraction is undefined. Since we can't have that, we have to find out what X values would make the denominator equal to zero. And to do that. We just take the denominator and actually set it equal to zero because that will give us our X values That make the denominator zero. So looking at X cubed plus X squared minus 49 X minus 49. We can factor this by grouping grouping X cubed plus X squared. We can factor out an X squared and then we have X plus one. Looking at -49 X -49. We can factor out a negative 49 and that will leave us with X plus one if that was going a little bit fast, definitely check out previous lesson videos on factoring by grouping. All right now we've got an X plus one as a common factor. So we can factor that out. So it's X plus one times X squared minus equals zero. We can bring down that X plus one. That's as factored as that's going to get But look X squared minus 49. That is a difference of squares. We recall from previous lessons that we can factor out the difference of squares by taking the square root of both terms. So the squared of x squared is going to be an X times an X in the first position of each one and taking the square root of that second one squared of 49 that's going to be a seven times a seven. And then by definition when we have a difference of squares, one will be plus and one will be minus. Well that is as factored as we can get it. So let's set each factor equal to zero. X plus one equals zero, X plus seven equals zero and X minus seven equals zero for x plus one equals zero will subtract one from both sides. And then we get X equals negative one. For X plus seven equals zero. Let's subtract seven from both sides. And then we get X equals negative seven for x minus seven equals zero. We're going to add seven to both sides and we get X equals seven. Okay, what did we just find again? We just went through and found three values that would make the denominator equal to zero. If we substituted them in for X, those three values can't be because then we'd have an undefined function. So the domain of X is X can be any real number except for those three. Now how do we express that domain? We can write that in interval notation. And so what we do is we start with negative infinity and we start as low as we can go. And we say X. Can be any number up until we hit what number up until we hit negative seven on the number line. And it just can't be negative seven. So we express that with the parentheses. Parentheses means nope can't be that one. We say that's in union with another parentheses. And starting at negative seven because it could be any number just after negative seven. All the way up to. Which number do we hit next on the number line, negative one. We close that with the parentheses because it can't be negative one and that's in union with and we say negative one because it could be any number just after negative one. All the way up to What do we bump into next seven. Close that with the parentheses because it can't be seven. Starting with the parentheses again. And starting with seven because it could be any number just after seven. All the way up to what we don't bump into anything else. So all the way up to infinity and we close infinity with a parentheses just as we started negative infinity with the parentheses because we can never quite obtain infinity. We look at our answer choices and this matches with answer choice D Well done. We'll catch you on the next one

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In Exercises 1–30, find the domain of each function. f(x) = (2x+7)/(x^3 - 5x^2 - 4x+20)