Easy Vs Hard Finding the Inverse

all right so in this video what we're going to do is we're going to solve two equations you can see I have two rational equations one is easy and one is hard and if you maybe have a teacher that was like me you might get this on your homework and then voila this is the one that shows up on the test but in reality both of these problems students make mistakes on that's why I wanted to include them because whatever it is about rational equations just the idea of having a fraction bar gets students confused so I'm going to work through them step by step you can see they're very similar but there's going to be some differences that I want to make sure I highlight for you so you don't have to or when they come up all right so in the first one notice I did y equals you know obviously we could do f ofx for y but again just to shorten the steps first thing you always do replace f ofx with Y now what we're going to do to be able to find the inverse is we need to go ahead and swap the variables okay so that's always the first step when you are trying to identify the inverse equation and then now what we're simply going to do is we need to solve for y now notice that when we have a rational equation or at least in a problem like these two we're going to have a variable in the denominator so we have to get rid of the variable in the denominator and the way to do that is to multiply by what is on the denominator now notice I'm not multiplying by what's on the denominator on the top in the bottom I'm only multiplying it in the numerator well because of the division property since this -3 is being multiplied by the y + 7 and I have a y + 7 in the denominator these we can say divide out they actually just divide to one but -3 divid 1 is just going to be-3 now over here we have y + 7 * an X and remember we need to solve for our y right so to solve for the Y we need to get rid of these parentheses well how do we get rid of these parentheses we need to apply the operation well what is the operation it's multiplication okay so now you can see we have something that's really not that bad now we just need to solve for y so it's like a typal literal equation here right we have YX plus 7x I can't combine them they're not like terms but I can use my inverse operations to solve for y I need to subtract a 7x on both sides and then go ahead and divide by an X okay so you can see I made a little mistake there I wasn't really paying attention you don't want to divide by y on both sides you want to divide by X right because what are we trying to do we're trying to solve for y now why put this little ne1 there or that is the notation we use to denote the inverse of this function so the function our equation y = -3 / x + 7 has a inverse y inverse of -7x - 3 / X now that was easy because in reality we're not doing too much you know there's nothing grounding you're just following you know getting the denominator off the bottom and then we got to just go ahead and use a little literal equation to solve using your inverse operations just make sure you put it in the right step now let's go and get to the hard because a lot of times students will understand this they take their notes on it they it they get it they're like okay I can do this for my homework and then like like I said earlier this shows up on a quiz or a test and like crap I don't know how to do this or what am I missing here so let's go and explore this because the process is going to be exactly the same first thing you always want to do is replace your X's with with your y's to be able to find the inverse equation or function I know why I said inverse equated we're looking for the inverse function but I'm just using Rd and Y again that could be F ofx you know for a problem you just swap f x with Y that just makes the math a little bit easier you could always keep it but it just makes doing this math a lot easier so just like we did over here we don't want y + 7 in the denominator so again I'm going to multiply by y + 7 on both sides okay those divide out I already explained that and then we're going to apply our distur of property so now we have a little bit of an issue over here this wasn't that bad because you just had to use your inverse operations to solve for y right subtract 7x divide by X and you're good to go over here we can't do that because we have a y on both sides well remember your algebra to one days right the first thing we learned about solving equations if you need to solve for a variable always got to make sure you get it to the same side so to do that what I'm going to have to do is subtract a y on both sides I'm going to solve for everything on the left hand side and then I'm going to subtract a 7x on both sides so therefore I'm going to get rid of the 7x over here and get rid of the Y over here okay so hopefully you kind of see whenever you subtract something something by itself obviously that's going to subtract a zero so we're going to leave it off but you got to make sure using the properties of quality whatever you do one side you have to do on the other so now I have two y's right I can't combine them like these X these ones X's can't combine them they're not like terms I can't combine these they're not like terms but I still need to solve for a y so this is the step right here this is what students forget how do I solve for y sometimes students will divide by y on both sides please for the love of mathematics do not do that we need to be able to figure out a way where we can take a term with two y's and able to get it rewrite it so therefore there's only one y showing and there's a process it starts with an F it rhymes with tractoring and it's coming to the top of my head I can't think about it ah yeah it's factoring right what do these two expressions share they're not right they're not the same we can't combine them but they both share a y right and don't forget factoring we can always fact out factor out our common elements since they both sh factor out share a y we can factor out a y when I do that I'm left with a x - one and now all I need to Simply do to be able to solve Y is divide by the expression X - 1