In Exercises 1–30, find the domain of each function. f(x) = 1/(x^...

Question

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

Accepted Answer

Welcome back. I'm so glad you're here. We have this function. We have F of X equals nine over X squared plus minus eight over X squared minus 81. And we are asked to find the domain. Well, how do you find the domain? Well you look and you see what's going on with your function. And in this case we have fractions and what could possibly restrict the domain of a fraction? Well, if the denominator is equal to zero then that would be undefined and we don't want it to be undefined. So we need to figure out what values of X make the fraction undefined. So we're going to set each of our denominators X squared plus 81 equal to zero and X squared minus 81 equal to zero. Then we'll find out what values of X would make a zero in the denominator and therefore having undefined fractions. So for the first one, x squared plus 81 equals zero. Let's subtract 81 from both sides. That leaves us with X squared Equals -81. We'll take the square root of both sides and on the left hand side of the square root of X squared believes us with X. And on the right hand side while the squirt of the negative will give us eye and the square root of 81 gives us nine. So we've got plus or minus nine. I the next equation, X squared minus 81 equals zero. Let's add 81 to both sides and that will give us X squared equals positive 81. We'll take the square root of both sides again And that will give us x equals plus or -9. Now what we're saying here is that if X equals negative nine I positive nine I negative nine or positive nine then we will have an undefined value. So X can't equal those numbers. Well nine I and negative nine I those are imaginary. So X is never going to equal those anyway on a number line for a function but it could run into nine and negative nine. So for the domain X could be all real numbers except for negative nine and positive nine. So in order to express that in interval notation, we're going to have a parentheses starting with negative infinity because it could be all those numbers from negative infinity up to negative nine but not including negative nine. So we put a parentheses and then union with another parentheses and starting with negative nine because again we can't include negative nine. But anything else All the way up to positive nine but not including that. So we use the parentheses in union with and again, parentheses to say again, starting at nine but not including nine. Come all the way up to positive infinity. And again a parentheses for infinity because we can't actually obtain infinity. We look at our answer choices and this matches with answer choice A Well done. We'll catch you on the next one

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