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Current time:0:00Total duration:9:44

Domain & range of inverse tangent function

Video transcript

we're told given G of X is equal to 10 of X minus 3 PI over 2 plus 6 find a G inverse of X and they want us to type that in here and then they also want us to figure out what is the domain of G inverse the domain of G inverse of X so I've got my little scratch pad here to try to work that through so let's let's figure out what G inverse of X is this is G of X so G inverse of X I'm essentially let me just write this is G of X right over here G of X is equal to tangent of X minus 3 PI over 2 plus 6 so G inverse of X I essentially can swap the I can replace the X with the G inverse of X and replace the G of X with an X so and then solve for G inverse of X so I could write that X is equal to tangent of G inverse of G inverse G inverse of X minus 3 PI over 2 plus plus 6 so let's just solve for G inverse of X I actually encourage you to pause this video and try to work through this out will work work it out on your own so let's subtract 6 from both sides 2 to at least get rid of this 6 here and so I'm left with X minus 6 is equal to the tangent of G inverse of X minus 3 PI over 2 now let's take the inverse tangent of both sides of this equation so the inverse tangent of the left-hand side is the inverse tangent of X minus 6 and on the right hand side the inverse tangent of tangent if we restrict the domain in the proper way and we'll talk about that a little bit it's just going to be what the input into the tangent function is so if you restrict the domain in the right way tangent the inverse tangent of the tangent of something of say theta is just going to be equal to theta once again if we restrict the domain if we restrict what the possible values of theta are in the way so let's just assume that we're doing that and so this is going the inverse tangent of the tan of this is going to be just this stuff right over here it's just going to be that it's going to be G inverse of X minus 3 PI over 2 and our in the homestretch to solve for G inverse of X we could just add 3 PI over 2 to both sides so we get and actually let me just swap both sides we get G inverse of X is equal to the inverse tangent of X minus 6 and then we're adding 3 PI over 2 to both sides so this side is now on that this sides now on this side so plus 3 PI over 2 plus 3 PI over 2 so let me let me actually type that a let me see if I can remember it so because I'm about to lose this on my screen so inverse tangent of X minus 6 plus 3 PI over 2 so let me write that down so let me type this so G inverse of X is going to be the inverse the inverse tangent so I could write it like this the inverse tangent of X minus 6 and yes it interpreted it correctly inverse tangent you can view that as arc tangent of X minus 6 plus 3 PI over 2 and it did interpret it correctly but then we have to think about what is the domain of G inverse what is the domain of G inverse of X so let's think about this a little bit more the domain of G inverse of X so let's just think about what tangent is doing so the tangent function if we imagine a unit circle if we imagine a unit circle so that's a unit circle right over there we can imagine it to be a unit circle my pen tool is acting up a little bit putting these little gaps and things but I think we can power through that so let's just say for the sake of argument that that's a unit circle that's the x-axis and that's the y-axis if you form an angle theta if you form some angle theta right over here the the tangent of theta is essentially the slope of this terminal ray of the angle or the the yeah I guess you clogged the terminal rate of the angle the angles formed by that ray and this ray along the positive x-axis so the tangent of theta is the slope right over there is the slope right over there and you can get a tangent of any theta except for a few so you can find the tangent of that you can find the slope there you could find the slope there you could also find the slope there you could find the slope there but the place where you can't find the slope is when this ray goes straight up is when this ray goes straight up or this ray goes straight down those are the cases where you can't find the slope there the slope you could say is approaching positive or negative infinity so the domain of tangent so tangents tangent domain so the domain is essentially all real numbers all reals except except multiples of pi over I guess you could say PI over 2 plus multiples of PI except PI over 2 plus multiples of PI where K could be any integer so you could also be subtracting PI because if you have PI over 2 if you add PI you go straight down here you add another PI you go up there if you subtract PI you go down here and subtract another PI you go over there so these are this is the domain but you can but given this domain you can get any real number so the range the range here is all all reals because you can get any slope here you can increase theta if you want a really high slope decrease state if you want a really negative slope right over there so you can really get to anything now when you're talking about the inverse tangent when you're talking about the inverse tangent by convention you're going to well took to make tangent invertible so that you don't have multiple elements of your domain all mapping to the same element of the range because for example this angle right over here has the exact same slope as as this angle right over here right over here and so if you have two Thetas mapping to the same tangent then that's not if you don't restrict your domain so that you only have one of them it's not going to be invertible and so the convention the convention is is that to make tangent invertible you restrict its domain so you restrict the domain restrict the domain to the interval from negative PI over 2 to PI over 2 in order to construct the inverse tangent so the inverse tangent you can input any any real number into it so the inverse tangent domain domain this is just a convention they could have picked they could have restricted tangents domains as long as for any theta it met or there's only one theta in that in that domain that maps to a specific element of the range but the the convention is well inverse tangent can or if the convention is to restrict tangent domain between negative PI over 2 and PI over 2 so invert tangents domain is all reals all reals but its range is restricted its range and this is by convention it's going to be between negative PI over 2 and PI over 2 and not including and not including them so let's go back to our original question right over here what is the domain of G inverse so let's look at let's look at our the domain of G inverse well G inverse the domain of this I could put any real number in here any any real number here now what this is going to pop out is going to be something between negative PI over 2 and PI over 2 but they're not asking us the range of G inverse that should have been a more interesting question they're asking us what's the domain of G inverse and I could put in any real number right here for X so let's put that in here so the domain of G inverse of X it's negative infinity to infinity but actually just just for fun and let's just verify that we got the question right and we did but just for fun actually I am cure let's just think about what the range of G inverse is so the range of this thing right over here is going to be between negative PI over 2 to PI over 2 that's for this part right over here and then we're going to add 3 PI over 2 s to it so the range for the entire function so the range for this thing the range is going to be what the low end if we add 3 PI over 2 to this this is going to give us 2 PI over 2 which is just going to be so 3 PI over 2 minus PI over 2 is going to be 2 PI over 2 which is just PI just PI all the way to 3 PI over 2 plus another PI over 2 is going to be 4 PI is over 2 or 2 pi or 2 pi so the range of G inverse of X is PI to 2 pi and and it's an open interval it doesn't include the boundaries but its domain you could put any value for x here and it will be defined