Let's think about whatfunctions yes, really do, and also then we'll think about the idea ofan station of a function. For this reason let's start with a prettystraightforward function. Let's speak f that x isequal to 2x to add 4. And also so if ns take f of 2, f the 2is going come be same to 2 times 2 to add 4, i m sorry is 4plus 4, which is 8. I can take f of 3, whichis 2 time 3 plus 4, i m sorry is equal to 10. 6 add to 4. Therefore let's think about itin a tiny bit much more of an abstract sense. For this reason there's a collection of things thatI have the right to input right into this function. Girlfriend might already befamiliar with that notion. It's the domain. The collection of all of the thingsthat I have the right to input into that function, that is the domain. And also in that domain, 2 issitting there, you have actually 3 over there, pretty much you couldinput any kind of real number right into this function. Therefore this is going to be allreal, but we're making the a nice contained set here justto assist you visualize it. Now, as soon as you apply thefunction, let's think about it means to take f of 2. We're typing a number, 2,and then the role is outputting the number 8. It is mapping us from 2 to 8. So let's do another collection hereof every one of the feasible values that my function can take on. And we can contact that the range. Over there are more formal ways totalk around this, and also there's a much much more rigorous discussionof this later on, particularly in the straight algebra playlist,but this is all the different values I have the right to take on. Therefore if ns take the number 2 fromour domain, i input it right into the function, we're gettingmapped come the number 8. Therefore let's let me attract that out. Therefore we're going indigenous 2 tothe number 8 ideal there. And also it's gift doneby the function. The role isdoing the mapping. That role is mappingus indigenous 2 come 8. This ideal here, thatis same to f that 2. Exact same idea. You begin with 3, 3 is beingmapped by the duty to 10. It's developing an association. The function is mappingus native 3 to 10. Now, this raises aninteresting question. Is there a method to get earlier from8 come the 2, or is over there a method to go earlier fromthe 10 come the 3? Or is there someother function? Is there some various other function,we can call that the station of f, that'll take us back? Is there some otherfunction that'll take us from 10 ago to 3? We'll contact that the inverseof f, and we'll use that as notation, and it'll takeus earlier from 10 to 3. Is there a means to execute that? will certainly that same inverse of f,will it take it us ago from-- if we apply 8 come it-- willthat take us earlier to 2? Now, every this seems veryabstract and difficult. What you'll discover is it'sactually an extremely easy to resolve for this train station of f, and also I thinkonce we fix for it, it'll make it clear whatI'm talking about. The the function takes youfrom 2 to 8, the inverse will take us ago from 8 to 2. So come think around that, let'sjust define-- let's simply say y is equal to f that x. So y is same to f that x,is same to 2x plus 4. For this reason I can write simply y is equalto 2x add to 4, and this once again, this is our function. You provide me an x,it'll offer me a y. However we want to walk theother means around. We want to offer youa y and also get one x. So every we have to do issolve for x in regards to y. Therefore let's perform that. If we subtract 4 native bothsides the this equation-- allow me move colors-- if we subtract4 indigenous both political parties of this equation, we gain y minus 4 isequal to 2x, and also then if we divide both political parties of thisequation by 2, we gain y end 2 minus 2-- 4 split by 2is 2-- is same to x. Or if we just want to write itthat way, we can just swap the sides, we obtain x is equal to1/2y-- same thing as y over 2-- minus 2. So what we have actually here isa function of y that offers us an x, i beg your pardon isexactly what we wanted. We desire a duty of thesevalues the map ago to one x. So we can contact this-- us couldsay that this is equal to-- I'll carry out it in the same color--this is equal to f inverse together a role of y. Or let me simply write ita tiny bit cleaner. We might say f inverse together afunction the y-- so we deserve to have 10 or 8-- so currently the selection isnow the domain for f inverse. F inverse together a function of yis same to 1/2y minus 2. For this reason all we did is we startedwith our initial function, y is same to 2x plus 4, wesolved for-- over here, we've addressed for y in terms of x--then we simply do a small bit of algebra, solve for x in termsof y, and we say that that is our inverse together a function of y. I m sorry is ideal over here. And also then, if we, you know, youcan speak this is-- you could replace the y v an a, a b,an x, everything you desire to do, so climate we deserve to justrename the y together x. Therefore if you put an x into thisfunction, friend would obtain f inverse of x is equalto 1/2x minus 2. So all you do, you deal with for x,and climate you swap the y and also the x, if you want todo it the way. That's the easiest wayto think about it. And also one thing I desire to pointout is what happens as soon as you graph the functionand the inverse. Therefore let me simply do alittle quick and dirty graph best here. And then I'll carry out a bunch ofexamples of actually fixing for inverses, however I reallyjust want to give you the basic idea. Role takes girlfriend from thedomain come the range, the inverse will certainly take girlfriend from thatpoint ago to the original value, if that exists. So if i were come graph these--just allow me attract a tiny coordinate axis appropriate here,draw a tiny bit of a coordinate axis ideal there. This an initial function, 2x to add 4,its y intercept is walk to it is in 1, 2, 3, 4, similar to that, andthen that slope will certainly look choose this. It has actually a steep of 2, so the willlook something like-- the graph will look-- let me do it alittle little neater 보다 that-- it'll look something prefer that. That's what thatfunction watch like. What walk thisfunction look at like? What walk the train station functionlook like, as a duty of x? mental we resolved for x,and then we swapped the x and the y, essentially. We could say now that y isequal come f train station of x. So we have a y-interceptof negative 2, 1, 2, and also now the slope is 1/2. The slope looks prefer this. Allow me check out if ns can draw it. The steep looks-- or the linelooks something like that. And also what's therelationship here? i mean, you know, this lookkind the related, it looks prefer they're reflectedabout something. It'll it is in a tiny bit moreclear what they're reflected about if we attract theline y is same to x. Therefore the line y equalsx looks prefer that. I'll perform it together a dotted line. And you can see, girlfriend havethe function and its inverse, they're reflect aboutthe heat y is equal to x. And hopefully, thatmakes sense here. Due to the fact that over here, onthis line, let's take an easy example. Our function, when you take0-- for this reason f that 0 is same to 4. Our role is mapping 0 come 4. The station function, ifyou take it f train station of 4, f station of 4 is same to 0. Or the inverse function ismapping united state from 4 to 0. Which is exactlywhat us expected. The function takes united state from thex come the y world, and also then we swap it, we were swappingthe x and also the y. We would certainly take the inverse. And also that's why it's reflectedaround y amounts to x. So this example that i justshowed you right here, duty takes girlfriend from 0 come 4-- maybe Ishould do that in the function color-- for this reason the role takesyou native 0 to 4, that's the duty f that 0 is 4, friend seethat appropriate there, so the goes native 0 to 4, and also thenthe inverse take away us ago from 4 to 0. For this reason f inverse bring away usback from 4 come 0. You experienced that ideal there.


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Once you advice 4 here,1/2 times 4 minus 2 is 0. The next couple of videos we'lldo a bunch of instances so you really understand exactly how to solvethese and also are may be to carry out the practice on ourapplication for this.