Consider the function g(x) = -2 csc (4x + π). What is the doma...

Question

Consider the function g(x) = -2 csc (4x + π). What is the domain of g? What is its range?

Accepted Answer

Hey, everyone in this problem, we're asked to find the domain and range of the function given below, we have that G of X is equal to negative six CC can of eight X plus pi hats. So we wanna think about this function, we're gonna start with the domain, right? And when we're doing the domain, sometimes it's easier to rewrite our CC can't function in terms of the sine function because the sine functions, it's just easier to remember. We often know what sine function looks like and where it's defined. And so let's rewrite our function. And we have that G of X is going to be negative six divided by s of eight X plus pi divided by two because we recall that KC can of X is equal to one divided by sine of X. It's the reciprocal. Now, when we write it like this, it's gonna be easier to see where our function is defined. And it's gonna be easier to find that domain. Remember that the domain is all of the X values for which our function is defined. If we write it in this way where we have a fraction, it's easier to pick out that denominator, look at where that denominator is equal to zero, that's gonna cause our function to be undefined. So we wanna exclude those points from our domain. And so in this case, our function is going to be undefined if sign of eight X plus pi divided by two is equal to zero. So we have s of something is equal to zero. Well, if we think about a sine curve and it starts at 00, it increases up to one decreases back to zero, down to negative one and back up to zero. And it crosses that axis at pi to pi and we can continue this on and on and on. So sine of something will be equal to zero if that's something which in this case is eight X plus pi divided by two is equal to some integer, multiple of pi. So in this case, we'll call it N pi where N is an intruder, right? All right. So we have this set of values now where our function is gonna be undefined. And what we wanna do is figure out what the X values are. And right now we have this equation that needs to be satisfied. We wanna figure out what those X values are. So now we have this equation and we need to satisfy sorry, rearrange and simplify for X, right. So we're gonna subtract pi divided by two on both sides, we get eight X is equal to NP minus pi divided by two. We can keep going. We're gonna divide both sides by eight. And we get that X is equal to NP minus pi divided by two, all divided by eight. OK. So these are the X values for which our function is undefined. These are the ones we wanna exclude from our domain. Now, we can simplify a little bit and rewrite this. We can write it as X is equal to N pi divided by eight minus pi divided by 16. Or if we wanted a common denominator, we could write this as two N pi minus pi all divided by 16. OK. So these are X values. And so if you wanna go ahead and actually write out what this domain is, remember, we're looking for the domain and range, we're gonna take all of the real numbers and we're gonna subtract these X values that give us no function by that, that the function is undefined for we have all the real numbers except these ones and these ones we're gonna write as N pi divided by eight minus pi divided by 60. OK. That is the domain for this function. OK. So we're done with the first part of the question. We can go to our answer choices and take a look at where we're at so far. Option A and option D have the same domain, right? And it's similar to what we found, but they're adding. So we have two N pi plus pi divided by 16. We had a subtraction. OK? So that's not quite the same. So we can eliminate option A and option D and then option B and C have the correct domain. It's written a little bit differently. OK. So we wrote it in two different ways. Option B has it as R minus N pi divided by eight minus pi divided by 60 A and option C has it written as two N pi minus pi all divided by 60. OK. Both of those are the same, we've shown writing both ways. And so either of those could be the correct answer. Now we move on to the second part of the question which is finding the range. So if we go and we think about the range, we're gonna start with side of X again, stick with what we know and work from there. OK? So sign of X, we know that the range is from negative 1 to 1. OK? If we just have a regular sine curve, that maximum value is one, the minimum value is negative one. And remember that the range is that range of possible Y values. So if we plug all of the possible X values into our domain, what are the possible Y values that we get? OK. And with sin of X, they're between negative one and one inclusive. And remember that Koi Can is the reciprocal. So if we take one and we divide it by something that's between negative one and one inclusive. And the thing that we get out of that is going to have a magnitude that is bigger than, or equal to one. OK? Because we're dividing by something that's either less than one or equal to one. What we get out of that is going to be bigger than one or equal to one in magnitude. And so the range of the regular Koc can't function RKOC can of X, we can write as negative infinity up to negative one inclusive we're gonna take the union with the interval, including one and up to infinity. OK. So again, we're just taking a unit of these two intervals and all this really is is taking all of the numbers that have a magnitude bigger than or equal to one. OK. So we know what the range is for the regular coast. You can't function when we just have X inside nothing being multiplied. So now we move to our function G of X and G of X, we know is equal to negative six KC can of eight X plus pi divided by two. Now this cosecant of eight X plus pi divided by two, we know has the range we just wrote down right. Now, our function is taking that and multiplying it by negative six. So we're taking something that has a magnitude of one or more. We're multiplying it by six. So we're always gonna get something that now has a magnitude of six or more. If we end up with one and we multiply it by negative six, we get negative six, we end up with negative one, we multiply it by negative six, we end up with six. OK. So those minimum values, the minimum magnitude of this function is actually going to be six now because of this constant multiplication outside. And so the range four G of X is going to be very similar to the range of KKX. But we're gonna have from negative infinity up to negative six inclusive. And we're taking the union with the interval from six up to infinity where we include that six. OK. All right. So this is the range of our function. That was part two of the question. We can go to the answer choices and compare what we found. We had narrowed it down to option B or option C. We can see that option C does not have the correct range. It has the range of just CC can of X, not our function G of X. Option B has the range we found. And so that is gonna be the correct answer. Option B, thanks everyone for watching. I hope this video helped see you in the next one.

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Consider the function g(x) = -2 csc (4x + π). What is the domain of g? What is its range?

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