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Current time:0:00Total duration:4:37

Video transcript

so we're told that H of X is equal to the negative cube root of 3x minus 6 plus 12 and what we want to figure out is what is the inverse of H so what is what is H inverse of X going to be equal to and like always pause the video and see if you could figure it out well in previous videos we we've emphasized that what an inverse does is it it will a function will map from a domain to a range and you could think of the inverse as mapping back from that point in the range to where you started from so one way to think about it is we want to come up with an expression that unwinds whatever this does so if we say that Y if we say that Y is equal to H of X well we could say that Y is equal to the negative of the cube root of 3x minus 6 plus 12 this gives us our Y and you could think of Y is a member of the range a member of the range in terms of what our input is in terms of a member of the domain but we want to go the other way around so what we could do is we could try to solve for X if we solve for X we're going to have some expression that's a function of Y we're going to have that being equal to X and so that would be the inverse mapping another way you could do that is that you could just swap x and y and then solve for y now that's a little bit less intuitive that this is actually the inverse so actually let's just solve for x here so the first thing that we might want to do is let's isolate this cube root on let's say the right hand side so let's subtract 12 from both sides and we would get Y minus 12 is equal to the cube root of it's actually the negative cube root don't want to lose track of that negative cube root of 3x minus 6 and then if we add we subtracted 12 from both sides so that 12 is now that 12 is now gone and now what we could do well we could multiply both sides by a negative 1 that might just get rid of this negative here so we multiply both sides by a negative 1 negative 1 and then we multiply this times a negative 1 on the left hand side well that's the same thing as 12 minus y and on the right hand side we're going to get the cube root of 3x - six and now and this is going to be a little bit algebraically hairy we wanted we want to cube both sides so let's do that so let's cube both sides and actually it doesn't get that algebraically hairy because I don't actually have to I don't have to figure out what this I don't have ticks I don't have to expand it I can just leave it as 12 minus y cubed and so if we cube both sides on the left hand side we're just left with 12 minus y cubed and on the right hand side well you take the cube of the cube root you're going to be left with what you originally had under the cube root sign I guess you could say and now we want to solve for X let's add 6 to both sides so we're going to get 12 minus y cubed plus 6 is equal to 3x now we could divide both sides by 3 and we're all done divide both sides by 3 and we get we get X is equal to 12 minus y to the third power plus the 6 over 3 and so this if you have a member of the right one way to think about if you have a member of the range Y this is going to map it back to the X that would have gotten you to that member of the range so this is the inverse function so we could write H inverse of Y is equal to this business 12 minus y cubed plus 6 over 3 and like we've said in previous videos this choice of is calling Y the input well it could be anything we could call that star we could say H inverse of star and we're just naming our input star is equal to 12 minus star cubed plus 6 over 3 or if we just want to call the input X we could just say H inverse of X and once again this is just this this is just what we're calling the input is equal to 12 minus x to the third plus 6 over 3 might be a little bit confusing because now in theory X could be considered a member of the range and we're mapping back to a member of the but either way we you know the quote we can we can call the we can call the input function to a function Brett practically anything but there you have it that is our inverse function that essentially unwinds what our original function does