hi it's Miss molsberry here and this video we are starting chapter 10 which is all about um inverse functions is one big topic we're going to talk about it today and radicals so different square roots cube roots um any radical we need uh which also means we're going to be talking about some exponent rules so this unit is actually our this is chapter 10 and then we have 11 and 12 and then we're done for the year so we're getting closer um you will have one weighted assessment over inverses in the third N9 weeks and then the rest of our grades will be in the fourth nine weeks for this uh unit so I'm going a little bit Rogue these notes that we're doing on the video don't match the paper exactly but it's close I'm going to write them on my own paper because I think it'll give you a better idea than if we just did the examples on the paper alone you're more than welcome to go back to the paper and fill it out with the key that's posted um so let's go ahead and talk about a power function we did last semester so those would be X raised to a different Power um and then a word that's going to be important we take it for granted but it's function So today we're going to be talking about functions again like Algebra 1 remember to be a function each X has to be unique the X can't can't repeat and we check from a graph by making sure that it passes the vertical line test as you probably remember from Algebra 1 and that's a quick way to check and make sure none of our X's uh repeat then we're going to start introducing a function's inverse so to find an inverse we are going to switch the X and the y in the equation and what that's going to do it's also going to make the domain and the range switch and there's a notation for inverse the notation that we're going to use I guess I should label it inverse notation instead of f ofx it's F ofx but with a -1 as like an exponent between the F and the X so we're going to start off by looking at a function you know well and then we'll talk about its inverse so the function we're going to start with is a power function it is f ofx = x^2 quadratic and we're going to make a tiny little sketch and then figure out what it's in inverse function looks like so if you think back to the fall which feels like a long time ago our parent function for um quadratic we had used the X's -2 through 2 and we know that when we Square them that would give us our y Valu since the equation is X2 and then I'm going to go ahead and draw a little sketch I promise you there's a point to this okay so the vertex at 0 0 -1 1 -24 positive2 4 oops okay there we go so that is a quadratic function let's talk about its inverse we're not going to talk about an equation equation yet but I'm going to use the inverse notation and we said that we switch X and Y so you can even do that from a table the X's will become the Y's the Y's will become the X's I'm going to switch colors as well so 4 -2 would now be an ordered pair 1 1 0 0 1 1 and 42 and then we're going to graph this and see what it looks like okay let's see now I need my xes to go out to four so it still has the0 0 1 1 um 4 -2 one one and 4 2 so if I connect this I can see that it is a sideways Parabola and I can also tell that this is not a function for two reasons let me write it first okay so not a function it would not pass the vertical line test if I drew a vertical line through this sideways graph I would reach the graph into two different places and I can clearly see from the table that the X's repeat each X does not go with a different y so these two examples they are inverse relations but not an inverse function because a function has to pass the vertical line test so here's what we're going to do let's kind of summarize a few things that we've t talked about so far um so we said this is not creating a function as an inverse one way that we can tell is by a new line test which is called the horizontal line test so we would perform it on the AR original function so here if we perform on a parabola a horizontal line test and it fails that tells me its inverse is not a function right so instead of having to go through the trouble of drawing the inverse I could have just sketched this graph done the horizontal line test and realized it was going to fail and in order to be invertible or what we call a one: one function I'm going to put one to one in parentheses there so those two are the same one: one or invertible those functions have to pass both the vertical line test that's not a very good V and the horizontal line test so obviously we now know quadratic graphs they do not pass the vertical and horizontal line test we'll talk about how we can make it have an inverse and be invertible in a minute but we've talked about some functions that are invertible so a few that you actually know linear a simple line is invertible we're going to practice finding the inverse of a line in a minute it passes the vertical and horizontal line test X SAR did not work but if we tried X Cub or any odd powered or odd degreed function think about what that looks like our cubic functions um that passes both the vertical and horizontal line test and then more recently from this semester one of our rational functions rational odd would technically also pass because it has pieces of its graph in quadrant one and three and it would pass both the horizontal and vertical line test okay so what are you really going to be asked to do let's kind of put all this vocabulary together how to find an inverse so I'm still going to ask you to find an inverse of a quadratic well how could that be well let's see we're going to look at only a piece of the graph so let me go ahead and get a new page just for the sake of space so how to find an inverse if our original function is f ofx = x^2 I have to provide you with what we call a domain restriction and this is what it looks like it will be an inequality telling you to only look at part of the graph so if we said the X's have to be greater than or equal to zero I'm going to use that table from the page above where it was 0 0 1 1 2 4 and only graph that part of the function okay so basically so far this is the right side only of the parabola we're not including any X's smaller than zero now I can find its inverse so now when I switch the X and Y so I'm still labeling them X and Y it's just that what was a y becomes an X oops we won't even really be able to tell here until we get to the third point 4 2 and then if we go and sketch this 0 0 1 1 42 it's only going to make the top half of the um of the parabola the sideways Parabola we were looking at and therefore this is okay this would pass the vertical line test and be a function okay so are we going to graph it every time the answer is no so let's talk about how we're going to find inverses and that's going to be the next thing we practice we're going to get to some more through examples the equation of an inverse is found by switching the X and Y and then solve for y so if we do that on a um the equation we just had y = x^2 first thing I'm going to do is switch the X and the Y so it's going to be X = y^2 and now I have to solve for y so I'm going to have to take the square root of both sides and when I do that remember typically that creates a positive and a negative square root but in our graph we are only talking about the positive X values which are creating this piece of our graph so um if you look at it what was the original X's of 012 if I go back up here it's only looking at 012 uh this piece of the graph the upper one is positive x^2 I'm going to use a highlighter for that so the positive only meaning I can just write I solved for y because a square root and square Square cancel I get the square root of x and so I can say the inverse equation is the square root of x it ends up having a domain so this is what our square root graph looks like it's one of the functions we're studying in this unit this function goes from0 to Infinity for both its domain and its range and we're going to have a transformation day uh next class okay one last thing before we find more inverse equations um over here one thing about the graphs the original F ofx and its inverse are reflections over the line Y = X so if you're curious if two graphs are inverses I'm going to plot the quadratic equation I'm going to plot the x s graph that had its restricted domain over here 0 0 1 1 2 4 and then I can see let me use a different color here this yellow if I was going to go plot the equation Y = X oops it's a little tricky to do this without a graph I should be able to see that each function is equidistant from the line yal X my graph is slightly less than ideal but I tried my best okay so I did restrict this one to X is greater than or equal equal to Z what if I restricted it the other way so let's do that really quick okay so if instead I said hey let's look at F ofx = x^2 but this time x has to be less than or equal to zero okay well let's go ahead and graph that so now I would only want your table to have 2 - 1 0 for your X's 4 1 and Zer for your y's and we're going to be looking at the left half of the graph there are those three points and that's it only the left side of this Parabola and so if I go to find its inverse if I'm using my same solving steps that we did just above the steps are going to look similar I'm going to switch the X and Y I'm going to take the square root of both sides but now this plus or minus we're going to actually be choosing the negative so the negative square root corresponds to the left side of the parabola so < TK X and then at the and it's always best to replace the Y with your inverse notation and state what you found as your inverse equation so we're going to go plot that and if it helps you can always just switch the X and the Y's so let me get those on here 4 -2 1 -1 0 [Music] 0 and you can see it's still a reflection across y = x the blue and the red are still the same distance uh from our linear parent function and I'll label which one's which okay man there's so much background info today and that's why I felt like the printed notes didn't do a very good job so y'all we kind of need to talk about one other type of function so uh we're going to have to restrict our domain anytime we have an even power because we know those have symmetry across the x or excuse me the Y AIS so what about if it's an odd power okay well let's go ahead and look at X cubed and we're just going to do a real quick sketch here so I'm going to put the same X's that we did oops for the quadratic graph but then remember we're cubing so -2 * -2 * -2 is going to be8 and then 1 0 1 and 8 so I'm not making my best graph ever here 0 0 there's my one one and negative 1 negative 1 and then two and I'm going to say this is 8 and -28 okay so here's my function X cubed and then if we notice it passes the vertical and horizontal line test so there's no need to restrict its domain to find its inverse we're just going to switch the X and the Y so to find the equation we're going to take X and set it equal to Y cubed and one of the things we're going to we talking about in this unit is that we can take any sort of root we need so instead of a square root we need to take a cube root to cancel our Cube y will now be by itself there's no need for a positive or negative when we're working with an odd function or odd power and so I'm getting the cube root of x which just means that these x's and Y are going to be switched so to help you graph it um I'm just going to switch the x's and the Y's and y'all this is one of the topics it's not super fun but I promise it comes back with a vengeance and pre-cal so you want to do your best right now to try to understand what's going on and if you don't understand like ask questions and know that it will get easier the more we practice um okay so notice these points are the same here but then I have 82 and8 -2 so the blue here is the inverse function and this is important because inverses inverse functions have different properties that are going to help us um they're going to help us as we do more complicated solving in some of the next units coming up so it is important to know the properties of an inverse function and how to find it uh one you need thing about our cubic and cube root function here we're going to transform these next class do shifting Reflections all that fun stuff but the domain and range are all real numbers or negative Infinity to positive Infinity for both the original function and its inverse okay so y'all there are two types of equations that you're going to be asked or two types of problems you're going to be asked to work out in this unit so you know what I'm probably going to make it a whole second video so this was all the background knowledge video and now I'm going to do a separate video where we actually work through some problems like the test or you know the Spiral Review so check up for video two