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let's think about what functions really do and then we'll think about the idea of an inverse of a function so let's start with a pretty straightforward function let's say I have f of X is equal to 2x plus 4 and so if I take F of 2 f of 2 is going to be equal to 2 times 2 plus 4 which is 4 plus 4 which is 8 I can take F of 3 F of 3 which is 2 times 3 plus 4 which is equal to 10 all right 6 plus 4 so let's think about it a little bit more of an abstract sense so there's a set of things that I can input into this function you might already be familiar with that notion it's the domain the set of all of the things that I could input into that function that is the domain that is the domain and in that domain 2 is sitting there too is there you have three over there pretty much you could input any real number into this function so this is going to be all reals but we're making it a nice contained set here just to help you visualize it now when you apply the function let's think about what it means to take F of 2 we're get-- we're inputting a number two and then the function is outputting the number 8 it is mapping us from 2 to 8 so let's make another set here of all of the possible values that my function can take on all of the possible values that my function can take on and we can call that the range there are more formal ways to talk about this and I there's a much more rigorous discussion of this later on especially in the linear algebra playlist but this is all the different values I can take on so if I take the if I take the number 2 from our domain I input it into the function we're getting mapped to the number 8 so let me draw that out so we're going from 2 to the number 8 right there and it's being done by the function the function is doing that mapping that function is mapping us from 2 to 8 this right here that is equal to F of 2 same idea you start with 3 3 is being mapped three is being mapped by the function to ten to ten it's creating an association the function is mapping us from three to ten now this raises an interesting question is there a way to get back from eight to the two or is there a way to go back from the ten to the three or is there some other function is there some other function we can call that the inverse of F that'll take us back is there some other function that'll take us from ten back to three we'll call that the inverse of F and we'll use that as a notation and it'll take us back from ten to three is there a way to do that is that what that same inverse of F will it take us back from if we take if we apply eight to it will that take us back to two will the inverse of F if we apply it to eight will it take us back to two now all of this seems very abstract and difficult but you'll find is it's actually very easy to solve for this inverse of F and I think once we solve for it'll make it clear what I'm talking about that the function takes you from two to eight the inverse will take us back from eight to two so to think about that let's just define let's just say y is equal to f of X so let's say y is equal to f of X so y is equal to f of X is equal to 2x plus 4 so I could write just Y is equal to 2x plus 4 and this once again this is our function it you give me an X it'll give me a y but we want to go the other way around we want to give you a Y and get an X so all we have to do is solve for X in terms of Y so let's do that if we subtract 4 from both sides of this equation let me switch colors if we switch if we subtract 4 from both sides of this equation we get Y minus 4 is equal to 2x and then if we divide both sides of this equation by 2 we get Y over 2 minus 2 right 4 divided by 2 is 2 is equal to X so if we just want to write it that way we can just swap the sides we get X is equal to 1/2 Y same thing is y over 2 minus 2 so what we have here is a function of Y that give an X which is exactly what we wanted we want a function of these values that map back to an X so we could call this we could say that this is equal to this is equal to I'll do it in that same color this is equal to f inverse as a function of Y or let me just write a little bit cleaner we could say f inverse as a function of Y so we can have ten or eight so now the range is now the domain for F inverse F inverse as a function of Y is equal to 1/2 Y minus 2 so all we did is we started with our original function y is equal to 2x plus 4 we solved for over here we have we've solved for Y in terms of X then we just do a little bit of algebra solve for X in terms of Y and we say that that is our our inverse as a function of Y which is right over here and then if we you know you could say this is you can replace the Y with an a a B and X whatever you want to do so then we can just we can just rename the Y is X so if you if you put an X into this function you would get f inverse of X is equal to 1/2 X minus 2 so all you do you solve for X and then you swap the Y and the X if you want to view it that way that's the easiest way to think about it and one thing I want to point out is what happens when you graph the function and the inverse so let me just do a little quick and dirty graph right here and then I'll do a bunch of examples of actually solving for inverses but I really just wanted to give you the general idea function takes you from the domain of the range the inverse will take you from that point back to the original value if it exists so if I were to graph these so let me draw a little coordinate axis right here draw a little bit of a coordinate axis right there this first function 2x plus 4 its y-intercept is going to be 1 2 3 4 just like that and then it will it's its slope will look like this it has a slope of 2 so it will look something like it's the graph will look let me make it a little bit neater than that it'll look something like that that's what that function looks like what does this function look like what does the inverse function look like as a function of X remember we solved for X and then we swapped the X and the y essentially we could say now that y is equal to f inverse of X so we have a a y-intercept of negative 2 1 2 and now the slope is 1/2 so the slope is 1/2 the slope looks like this let me see if I can draw it the slope looks or the line looks something like that and what's the relationship here I mean you know these look kind of related it looks like they're reflected about something and it'll be a little bit more clear what their reflected about if we draw the line y is equal to X so the line y equals x looks like that I'll do it as a dotted line right the line y equals x looks something like that that is the line y is equal to X and you can see you have the function and it's inverse they're reflected about the line y is equal to X and hopefully that makes sense here because over here on this line let's take an easy example our function when you take when you take 0 so f of 0 is equal to 4 or our function is mapping 0 to 4 right the inverse function the inverse function if you take F inverse of 4 f inverse of 4 is equal to 0 or the inverse function is mapping us from 4 to 0 which is exactly what we expected the function gives us takes us from the X to the Y world and then you swap it we were swapping the X and the y when we take the inverse and that's why it's reflected around y equals x so this example that I just showed you right here function takes you from 0 to 4 they actually do that in the function color so the function takes you from 0 to 4 that's the function f of 0 is 4 you see that right there so it goes from 0 to 4 and then the inverse takes us back from 4 to 0 so f inverse takes us back from 4 to 0 you saw that right there when you evaluate 4 here 1/2 four minus two is zero the next couple of videos will do a bunch of examples so you really understand how to solve these and able to do the exercises on on our on our application for this