Functions & Graphs – Grade 11 & 12 Maths

today hello mindset says welcome to it it is a Monday and it’s Matt’s for great halls at 6 o’clock I am uni and joining me in studio is Lisl house the two owls tonight I’m very well I’m very well and I’m really looking forward to this lesson on exponential and log rafts inverses and so forth alright guys so she told you what we’re doing so Lisa I’ll send you to your board while I tell the mines it is what you need to know for the show so guys you can contact me on facebook at facebook.com/ foot slash learn extra you can also hit me up on twitter it’s been very quiet lately so guys please talk to me until they’re at then extra you guys can download all your show notes at learn extra Roxio today for /live all your videos all your schedules will be there on that page and guys I’ll be telling you about future stars later on don’t forget it is Earth Day so hope you guys doing good for the earth you guys not polluting anything and you guys doing great stuff no but anyway guys I hope you’re ready for the show because you about to kick start with leaves also Lisl take it away right guys so we’re gonna start the show by just quickly revising your basic exponential graphs from grade 11 because the inverse of these exponential graphs are log graphs so if you don’t know what the exponential graphs look like you’re going to have a bit of trouble with the law graphs so before we have a look at the exponential graphs itself I think it’s very important that we just talk about some terminology now the first thing that I want to talk about today is what is a function a function is a mathematical rule that map’s an input value to a unique output value now guys we have dealt with functions before exponential functions exponential graphs of functions hyperbolas are functions parabolas or functions straight line graphs of functions so basically a function is a rule that that map’s an input value to a unique output value now one quick way to check whether you are dealing with a function is to use something called the vertical line test now let me bring up any graph let’s do a parabola right so I’ve got a parabola over there now I know that a parabola is a function but if you want to test using the vertical line test what you will do is move your ruler in such a way that at any given time you are parallel to the y-axis so let’s just do that so so at any given time you parallel to the y-axis and what needs to happen in order for this graph to be a function is that your ruler can only cut your graph once at any given time what do I mean by that well if you have a look at this graph over here let’s bring up a ruler can you guys see that and my line won’t work isn’t that lovely can you guys see that here the vertical line test is not passed because that vertical line cuts our graph in more than one place so the easiest way to check whether we are dealing with a function is the vertical line test and an et given place when I draw a line I’m only allowed to intersect once right so here are the notes and if you downloaded the notes you’ll have them we use a ruler to perform the vertical line test on a graph to see whether it’s a function or not hold a clear plastic ruler parallel to that y-axis in other words vertical move it from left to right over the axis if the ruler only ever cut cuts the curve in one place throughout the movement from left to right then the graph is a function right so that’s very very important right some other things that we need to talk about are domain and range now guys by the time you get to grade 12 you should really know what domain and range are because you’ve been doing this since grade 10 now domain is the input or the x value so domain is

what is happening here what X values is my graph defined for where as the range is the Y value so to give you guys a quick example let’s say I have again a parabola and I tell you that that parabola has a maximum turning point at X is 1 and Y is 4 if I now ask you for the domain and the range of that parabola you need to investigate now let’s look at the x values is there any restriction at all on the x values of this graph no it will carry on forever and ever and ever to positive infinity and negative infinity so we can say that the domain is X an element of the real numbers alternatively you can use this notation where you say X is an element of negative infinity to infinity and remember that the infinity signs are always accompanied by round brackets now if we look at the range of this graph we look at the Y values that this graph is drawn for so the highest Y value that this graph ever gets to is over here at 4 so it goes from negative infinity up towards 4 but then it starts going down again so we’re not talked about the range of this graph I will say Y an element of negative infinity all the way up to four or alternatively I can say why and actually it’s included at four sorry so that needs to be a square bracket because my graph is drawn at four or I could say Y is less than or equal to four these two notations are equivalent so usually guys you’ll have the sketch in front of you you need to ask yourself just some simple questions what’s happening with the X’s and what’s happening with the Y’s right so that is our domain and range we’ve spoken about the vertical line test now before we carry on I just quickly need to talk to you guys about your basic exponential graphs right so the general equation of an exponential graph is y is equal to a B to the power X plus P plus Q now guys most of the time our graphs won’t actually be that intricate but what does the Q do the Q facilitates the up and down shift of my graph and the P facilitates the left-to-right shift of my graph and then the a governs the steepness right so how quickly does it go uphill now I’m going to start by drawing a very basic exponential graph y is equal to 2 to the power x now guys I would recommend that you know all the properties for these basic graphs because everything else you will you will deduce from your basic graph now you can use on your calculator you can use your table function if you’re not sure of what that graph looks like in case you guys don’t know how to use that mode and I’m going to go three for table and then I’m going to say f of X is equal to 2 to the power and alpha X so there I’ve drawn a in f of X is 2 to the power X now guys you should know what your graphs look like but if you’re unsure then this is one way to deal with it then it’s going to ask you where do you want to start now when I asked about the start we are talking about our input values or our X values so let’s start at minus 4 and where do we want to end I think 5 is sufficient and we’re going to go up in steps of 1 then what you grab what your calculator is going to do for you it’s going to give you your input or your x-values which will be your domain and it’s also going to give you the corresponding output or Y values which is your range now let’s have a look here quickly at minus 4 the Y value is very very small at minus 3 it’s also small but it’s getting bigger and as I carry on I can see that at X is minus 1 Y is 1/2 at x is 0 Y is 1 and what then starts happening is the graph increases

very rapidly so we need to know for a graph in the form Y is equal to a to the power X as long as a now that’s in this case my 2 is bigger than 1 so let’s just write there a is bigger than 1 what your graph is going to look like is like this it’s going to get very very close to the x axis and then it’s going to increase quite rapidly that’s the basic curve that we’re looking at that y-intercept over here is at x is 0 Y is 1 now in the case of this basic graph you’re you have got a horizontal asymptote and that is your x-axis and the reason for that is we can see this as Y is equal to 2 to the power x plus 0 if I were to now do Y is equal to 2 to the power x plus 1 that will vertically shift my graph up by one unit so instead of my asymptotes being the x-axis or the line y is equal to zero it will now be Y is equal to one and everything else will shift from there so it’s very important that we know what our basic graph looks like that if I have a look at Y is equal to 2 to the power x I can see that the domain the X values is X er there is no restriction on the X values and the y values or the range is from the horizontal asymptote upwards and our in this case it will be from 0 to infinity so that is our very very basic exponential graph I’m going to do the other one as well so let’s just I’m just going to make a quick summary we’ll say this is y is a to the X where a is greater than 1 there’s my y intercept at 1 and my domain is X and element of the real numbers and my range is y greater than I’m going to put H a for horizontal asymptote but in this case it’s 0 right then we can also draw a graph y is a to the power X where a is not bigger than 1 where a is greater than 0 but less than 1 so that would be something like Y is 1/2 to the power X now the graph wires are 1/2 to the power X is exactly the other way around from its friend over here there are reflections in the y-axis so this the graph y as a to the X where a is bigger than 1 is increasing so as I look from left to right the graph gets bigger and bigger and bigger this one is decreasing so it’s exactly the other way around but again the domain is X an element of the real numbers and Y is greater than the horizontal asymptote and in the case the horizontal asymptote is at zero all right so these are the basic basic properties of the exponential graphs that you have to know now I’m going to talk to you guys about what is an inverse now in particular the inverse of exponential graphs get asked a lot but you can get the inverse of any graph so let’s have a look it says here and inverse is a function an inverse of a function is a mapping of all the output values to the input values so basically what we’re saying is we are reflecting our graph in the line y is equal to X so that the original function has a certain set of x and y values for the inverse what we do is we swap the X and the y values around and for that reason my graph is that in your your function and its inverse will always have symmetry in the line y is equal to X now let’s have a look at the first graph that we drew I’m going to draw Y is equal to 2 to the power x on here right so that’s why is equal to two to the PI X now the inverse of Y is equal to two to the power X is a graph such that if I were to draw this in a very wet Cokie and fold on that line y is equal to X that would be the resulting

graph so without knowing even what it looks like yet or without knowing the notation even the graph will look something like this so that if I fold onto that line if I fold onto that line the two graphs will be on top of each other all right so just quickly before we carry on any further what happens when I do an inverse let me start with given F where F is the graph Y is 2x plus 1 okay so this is my function my original function now I’m going to have a look at the inverse of this function now what did I say just now I said an inverse is a reflection in the line y is equal to X so actually what happens is x and y trade places so instead of having Y is 2x plus 1 the inverse will be X is 2 y plus 1 and can you guys see that they have traded places my X is now where the Y was and using another color the Y and the X are also switched around now usually and it is considered more elegant to write our graph in the form y equals so what I’m going to do here I’m just going to make wire the subject of the equation yet again and this is going to give me Y is equal to 1/2 X minus 1/2 and if you were to draw this graph and this graph you will see that they are symmetrical in the line or Y is equal to X well it’s already time for our first break I feel like I’ve been talking nonstop Ludi what’s happening on the page are people following yes everyone’s enjoying it everyone’s tuned in so yeah fantastic guys we’re going to give you a quick break and then we’ll be right back here and we’re going to start looking at some actual exam type questions on functions and inverses yes guys you are going to take a very short break I just like to let you know about the future stars competition enter that guys they’re so many prizes to it the information is on our Facebook page so if you just go through that see what you need to do ask your friends to vote for you and then just click on it and look through it and just you know yeah guys I also like to remind you that it is Earth Day so just be safe you know keep this Earth cool don’t waste water don’t waste electricity all of that you know mother earth is a great place to live and we don’t want to live anywhere else guys so do take care of it we’re going to take a very short break so do stay tuned and we’ll be back after this we’re back guys I hope you’re fresh I hope you got some more time and some juice and you’re ready to continue with the lesson so I won’t waste any time and I’ll take it back to my beautiful teacher all right so thank you for that okay guys so we’ve established now that to get an inverse what we do is we swap the X and the y around now I want us to go back to our basic exponential graph let’s talk about Y is two to the power x right so we’ve said earlier on we know what this is the line y is equal to X we know what this graph looks like it increases very steeply from left to right like so okay and we also know that the graph y is equal to X or the function y is equal to X its domain is X an element of the real numbers in other words all X’s and y greater than zero okay now let’s talk about the inverse of this graph I’ve shown it to you guys before we said that the inverse will be a reflection in the line y is equal to X now just now when I was dealing with the straight line graph and its inverse what I did to get that inverse is I swapped the X and the y values around and then I made Y the subject of the equation yet again now have a look what happens when I do that over here instead of Y is equal to Y is equal to 2 to the power X I’m going to write X is 2 to the power Y alright that’s fine so we know that the equation of this graph is x is 2 to the power Y

however guys it’s not conventional for us to write our own set our equation as x equals we always have to write it as y equals and it is because of this little tricky situation over here that log notation was invented now there’s a process involved in making wire the subject of the equation you don’t have to ever show this but I’m going to quickly go through it basically what we’ll do is we’ll insert a log on either side then we will use the log law that says um if I’ve got log of B to the power a I can write it as a log B so basically that allows me to bring that Y in front so what I’m going to have here is I’m going to have Y log 2 is equal to log X and then dividing both sides by log 2 I get Y is log to the base 2 of X now if we have a look here it’s actually very easy if you have as your original graph your original function y is equal to 2 to the power x its inverse will be y is log to the base 2 of X so you can just learn that whatever it now appears over here will appear over here and the reason why the log notation was invented is so that we can make y the subject of the equation guys let’s just quickly have a look i’ve spoken about the domain and the range of my exponential graph the purple one now let’s have a look at the domain and range of my log graph what has now happened here is for the domain do you guys see that this graph is not defined for negative x values it stops right here at the y axis so the domain is X is greater than 0 however the range it can go up for as long as it wants to and down for as long as it’s once 2 is ye are now what I want you guys to notice is that my original graphs domain becomes the range of the inverse because what happens with inverses the X and the y values switch around so that’s really really important just a couple of things and if I look at the graph y is 2 to the power X it’s very definite that y is equal to 2 to the power x just the purple graph on its own is a function and the reason for that is because it passes the vertical line test its inverse is also a function because the inverse passes the vertical line test as well right we’re going to have a look at some point later on at when the inverse of a function is not a function but that’s more stuff for next week right so now just to consolidate whether you guys know what’s going on whether you can do your basic graphs we’re going to move on to the questions that I’ve selected for today these questions all come from past papers so if you can do them you can feel quite confident that you know more or less what’s going on so let’s look at those questions okay on your notes if you’ve done loaded them like you should you will get some notes on the log and the exponential function which I have explained just now okay the quest question says given that f of X is equal to 1/3 to the power x determine if to the minus 1 writing your answer in this form so whenever we see that notation F minus 1 we are looking for the inverse now this particular question doesn’t require us to do a sketch but because we’re still getting used to what everything looks like I’m going to start out before I even bother answering the first question by just drawing a rough sketch of f of X is 1/3 to the power X and this you should know from the properties of the graphs that I am discussed just now remember I said for a graph in the form Y is equal to a to the power x if a is greater than 1 your graph will be increasing from left to

right looking something like that for a graph in the form Y is a to the power X if a is greater than 0 but less than 1 in other words a fraction like what I’ve got here Y is equal to 1/3 to the power X then that graph will be a reflection in the Y axis of the other graph so this graph that I’m looking at is going to be decreasing right and I know that the y-intercept is where X is equal to zero and for a third to the power 0 is 1 so that’s basically what my graph looks like ok so that wasn’t required but I’m just doing that so that you guys can follow what’s going on the first question I’m asked is to give the inverse now I’ve started with an exponential graph its inverse will be a log graph fine so if my exponential graph is 1/3 to the power X then it’s inverse is going to be and we’ll write it over here y is equal to log to the base 1/3 of X right take careful notice my general formula is if I’ve got Y is a to the X its inverse will be Y is equal to log to the base a of X I have a look at the place swapping me right next question deals with the domain and the range of my inverse graph now you don’t actually have to draw the inverse graph to know the domain and range because you know that the domain and the range will swap around what do I mean by that these two graphs are symmetrical in the line y is equal to X so if my original graph f had a domain X er then my inverse will have a range of Y and elements are the real numbers I’ve drawn it in purple but you don’t actually even need to draw it you can just know that from the properties of the original graph and then again M if my original graph had a range of Y is greater than zero then the domain of my inverse graph will be X is greater than zero so yes I’ve drawn it to show you guys but it’s very important that you know the domain and range of your basic exponential graphs because you know for the inverse they just swap around Looney how’s the page looking now people thinking enjoying the show sir wonderful 100% then so let’s get on with the rest of the questions right so question C says give the equation of the line about which F and F minus 1 it’s inverse are symmetric now guys we spoken about this at length any inverse and it needn’t be a log in an exponential graph are symmetrical in the line y is equal to X so if we have a look earlier on when I was still doing nice of pictures exponential graph log graph and the symmetry most definitely lies in the line y is equal to x and that’s not just true for log and exponential graphs log and exponential graphs are inverses of each other but for any graph a straight line and it’s inverse a parabola and its inverse the symmetry will be in that line y is equal to x so if you were to fold the page on the line y is equal to x then the two graphs will fit on to each other perfectly right let’s have a look at the rest of this question let’s get there right the final question says determine G the reflection of F in the y-axis okay so let’s have a look if is my original graph and that was a third to the power X now in our lesson last week we spoke

about reflections and we said that if we reflecting in the y axis what happens the x value becomes negative and remember you don’t have to commit this to memory make yourself a little point say X is 1 Y is 3 I now want to reflect in the y axis that’s over here minus 1 and 3 will give me that reflected point in other words the Y value stays the same but the x value changes its sign so for the reflection and the y axis I will say f of X is 1/3 and because the x value becomes negative negative x ok guys one thing at this point which i think is really important and a lot of students don’t always realize this is I’ve been talking about 1/3 to the power x for this questioner there was a bracket over there that’s 1/3 to the power x please note that instead of 1/3 I could write 3 to the minus 1 all of that to the power X so in other words 3 to the minus X so please notice that 1/3 to the power x + 3 to the power negative x is in fact the same thing but anyway and so the in the equation of the graph reflected in the y axis is f of X equal to 1/3 to the power minus X now this 2 I can simplify I can say 3 to the minus 1 everything to the power minus X so I get three to the power X and guys very early on this program I showed that to you guys I said that this is my increasing graph that was two to the PI X now it’s reflection in the y-axis will be 1/2 to the power X or 2 to the power negative X I’ll hope that you guys are following the main thing for me is with most of these sorts of questions after a while they do all start looking the same so it is really important that you get enough practice now let’s just have a quick look at what our next question is about right so in our next question we’ve modeled part of the FIFA World Cup logo onto an exponential graph you guys will remember what if you look back now at papers from 2010 you’ll see everything is themed around soccer so what we have here is an exponential graph and in the form f of X and equal to a to the power X this graphs domain is restricted so this graph is only drawn from X as minus 2 to X is 3 so let’s say it finishes there at X is 3 and it starts here at X is minus 2 it’s fine usually an exponential graphs domain will be X er but we can restrict the domain that’s not a problem right we are also told that 2 + 4 is a point on our curve and then we are asked various different questions so I think I’m loony what we’re going to do is we’re going to take a bit of a break while our students think about this question and then we’ll be back and answering it ok guys as he’s also just think about it for a few seconds and you’re going to take a short break I’d also like to remind you guys that if you have any questions and we don’t get to them you can write your question to help desk learn extra dicey odors a and I teach how well I’ll see you guys within like 48 hours so don’t sweat it if you have a question and we don’t get to it during the show just go to that website and type your question in B and then you’ll get help don’t forget to hit me up on Tyler Oakley an extra on facebook at facebook.com/ for slash learn extra and you can download all your show notes the videos the schedules unlearn extra Co today forward slash life right now though we are going to take a very short break so do stay tuned we’re back guys will hook your first I hope you’re ready to carry on with the questions I hope you thought about the question that you’re ready to start again so Lisa take it away right guys so before the break we said we had an exponential graph modeled on the FIFA World Cup and it is f of X equal to a to

the power x my domain is restricted from an including -2 to and including 3 but we’ll deal with that in a minute the first question asks us to find the value of a now it is very important to have some sort of an idea of what you’re working towards now I can see that my graph is increasing and I know that my exponential graph is only increasing if a is bigger than 1 so that at least gives me some sort of an idea of what I’m working towards analogies know if I made a bit of a mess up so I’m going to start by saying y is equal to a to the power x then I need a point on the graph which I do have X is 2 and Y is 4 to substitute in here so I’m going to say 4 is equal to a to the power 2 so if a squared is 4 then a must be 2 so very very simple calculation you write down your defining equation they gave it to us and you need one point x value and a y value an ordered pair that lies on the graph and you can substitute in to find a so pretty straightforward right now let’s have a look at the next question and this is where you guys will start seeing how things form a pattern now they say find the equation of G where G is a reflection about the line y is equal to X so what are they asking us for they are asking us for the inverse right so at this point we should know that if we start with a to the power x our inverse is y is equal to log to the base a of X right so I’ve shown you guys what we do to get from the one form to the other we basically swap the X and the y value around and then we do a little log manipulation but it’s perfectly sufficient for you to now just use this fact that I’ve written down here for you so therefore if my original graph is y is 2 to the power X its inverse G will be log to the base 2 of X right so I’ve just applied this little rule ok now they start asking us about the domain of my inverse graph so let’s go back to our original graph for my original graph the domain because it was restricted by the question the domain was X an element of negative 2 up to and including 3 that was my domain and for my original graph the range was y greater than zero right now remember that my domain and my range now swap around because the two graphs are inverses of each other X becomes y and y becomes X so if I now look at the domain of my original graph that will now be the reign of that so if I look at the range of my original graph that will be the domain of my inverse so this will be X an element of and it will have to be 0 to infinity but just to fix here quickly I’ll need a round bracket at the 0 and I’m sorry a round bracket at infinity as well the reason I need a round bracket is because X won’t actually get to 0 because the y-axis or the line X is equal to 0 will be my asymptote right so let’s carry on and have a look at the rest of our questions now they say gee so this is this guy over here that I’m blocking off G is now reflected in the x-axis to give me a graph H right now we’ve spoken last weekend earlier today about what happens if we reflect in the X and the y axis if I reflect on the y axis X becomes negative if I reflect in the x axis Y becomes negative so now I

need to give the equation of this line so basically what I’m doing is I’m putting a minus in on the left hand side here but because we want our goal from the form y equals the – now goes there so that’s minus log to the base 2 of X right so we just keep on reflecting our growth not next question asks me find the value of H of one okay now what that means is in the place of X what I need to do is I need to replace the X with the 1 so I need to find the value of log to the base 2 of 1 now I know what the answer is but particularly because we haven’t sketched the graph this is where our calculator comes in really handy and let’s put it let’s put up let’s do a whole calculator I’m going to say f of X is equal and I’m going to do that I want to put a minus in front I don’t know actually how I’m going to do that okay but I’m going to put a 2 over there and a 1 over there oops oh yeah it’s not my lucky day today f of X is – let’s do it like that and log + 2 scrolling along – 1 equals okay now start at minus 4 that’s fine in fun step 1 the main thing that I’m looking for now is I want to see what is the Y value at X equal to what and so I’m just going to scroll down and at X equal to 1 the Y value is 0 now guys you can learn for yourself that the log of 1 it doesn’t matter what its bases it absolutely doesn’t matter what its bases the log of 1 will always be 0 but I’ve just shown it to you guys now by using the table function on my calculator right let’s have a look at the next question the next question says find the value of x if H of X is equal to 2 now I’m going to leave the calculator out of this now I’m going to say write H of X is minus log to the base 2 of X and basically I’m now going to say – log to the base 2 of X is equal to 2 right so that’s what I’m going to do and now putting this what I’m going to do is I’m just going to take the minus back to the top I’m allowed to do that so log to the base 2 of X to the power minus 1 is equal to 2 now if I put this from Log form into exponential form I get that X to the power minus 1 is 2 to the power 2 now we’ve spoken about log and exponential form before so X to the minus 1 is 4 so basically 1 over X is 4 over 1 and then cross multiplying and solving for x there are easier ways to get there X is equal to 1/4 so basically in the last two questions what they did is they gave us an x value they asked us for the y so they gave us an input they asked us for the output and then they did the opposite they gave us the output and they asked us for the input right so I hope that you guys are following and louny and any questions comments or suggestions from the page was everybody managing to follow quite okay yes and yes I think I’m says I say some problems and finding the gradient on finding the gradient that’s not really applicable now with functions what she might be referring to is more calculus where you have to get the derivative and substituting in the x value to find the gradient but in this particular section boys and girls we’re not really dealing with the gradient much okay right and then gamma is asking what do you mean by a restricted domain okay what do I mean by restricted domain that is a very good question what I mean by that is usually for an exponential graph the graph is drawn for X and element of R in other words it goes all the way to negative infinity and all the way up to positive infinity but in this question in particular they told me it’s only drawn from minus 2 to 3 so they gave me the restricted restriction right if they write nothing then there is no restriction but if they specifically give you a restriction on the domain then you must use that restriction on the domain and remember that the restriction on the

domain in the original graph will be and you know the X and the y will swap around for the inverse so the restriction on the domain is very much something that they have to give you they’ve got to put it there in brackets right so can we move on to our next question yes okay they say to us consider the functions f of X is equal to 2 to the power a 2 times x squared and G of X is 1/2 to the power X now this over here what we’re dealing with here is a parabola now we definitely going to do more of these sorts of questions next week this graph over here is an exponential now guys my feeling is always regardless of whether I’m asked to draw the graph or not when I read the question I have a mental picture of what the graph looks like the same way if you read a book you’ve got a mental picture of what your hero looks like my heroes are parabolas and exponential graphs lined up like to know what they look like if I look at f of X equal to 2x squared that is a parabola it will be symmetric around the X around the y axis and it’ll look something like this it’ll turn there at 0 0 ok now the exact steepness you won’t know but if you don’t have no idea what the graph looks like remember to use the table function on your calculator so this is f of X equal to 2x squared fine now over here I’ve got an exponential graph and the a part the base is 1/2 and we have learned tonight or I’ve taught you I don’t know whether you’ve learned that if I’ve got a fraction there between 0 and 1 my graph is decreasing right so it’s going to look something like that fine and that is G of X and that is 1/2 to the power X fine put a bracket there let’s have a look at our questions so the questions make a lot more sense if you know what you’re talking about that’s really really important so the question says to me firstly restrict the domain of F so that the inverse of F will also be a function right now because we haven’t really done this before this is more teaser for next week I’m going to draw the inverse of F for you guys remember inverses are symmetrical about the line y is equal to X so if I were to draw this graphs inverse it’s going to look a little bit something like that ok like the purple graph now the original graph my green graph was a function because it passed the vertical line test however if I now hold my ruler over the purple graph I can see that it intersects in two places so therefore the inverse is not a function now they say I must restrict the domain of F so that the inverse is also a function now do you guys agree with me if one of these hobs of these purple halves weren’t here then the inverse would still be a function so let’s have a look if my original graph looked like that only then it’s inverse would look like that and the original graph and its inverse would be a function similarly if my original graph only had that leg like that over there then the inverse would be something like that okay so basically what do I need to see here is as long as I’ve got both arms of Matt Farah Bella its inverse will fail the vertical line test so it’s inverse won’t be a function so I now need to give an instruction to make the original graph only half of the graph now I said to you guys earlier that this parabola will turn at zero zero so I’ve got two choices I can either restrict the domain to only the negative x values and I’m talking about the restriction the restriction on the domain is now the original graph so I can say X an element of negative infinity to zero then my original graph

will only have that on and its inverse will only have that on that’s my one option you get one of two options alternatively I could go to this option where only the positive arm is drawn so the inverse will be also only one arm and this is then X and element of 0 to infinity right but guys more about that next week but the point is if you have a parabola which is a one-to-many function its inverse won’t be a function so you’ve got to take away either the one side of the parabola or the other side in order to make its inverse still a function right just quickly let’s have a look at the next part of the question and I have it I just want to see here quickly we haven’t actually really been asked anything about the exponential war for anyway the exponential graph um its inverse will still be a function right the next question says the inverse oh sorry restricted our main of F so that the inverse is also a function and then that should be a and this should be B B says write the inverse of G in the form G to the -1 x equals now this is what I’ve taught you guys earlier tonight this will be its inverse will be loved to the base 1/2 of X so remember if the original is y is equal to a to the power X its inverse will be log to the base a of X so they have answered that question fine then there’s yet another question question C and the question C says the inverse of a function is 2x minus 4 give the equation of the original function so guys to move from a function and it’s inverse interchangeably if my inverse is 2x minus 4 how did I get to 2x minus 4 from the original graph by swapping the x on the way around so I’m going to say okay my inverse is 2x minus 4 but to get back to the original graph I’m going to swap the X on the way around so I’m going to say X is equal to 2y minus 4 and remember what I said to you guys earlier on in the program it’s considered really ugly to write your answer not in the form y equals so I’m going to do that bring the 4 over and then Y is 1/2 X plus 2 and what you’ll see in this one if you swap x and y around you’ll get to this one and vice versa loonie time flies when you’re having fun either start a goodbye already and we’ve got a whole lot of questioning guys all good things have to come to an end thank you so much for welcoming us into your homes for commenting on the Facebook page for tweeting us everything that you needed to do throughout the show guys it’s been amazing it’s been great thank you so much for tuning in and I just like to remind you guys that she’s happy you can download all the show notes the schedules and the videos on our learn extra does Co the delay for / lab page so you can do all that guys during the holidays drink the week during the weekend you can always go through that page guys and you see what you needs to get from the information that you’re given on that page thank you so much for joining us guys and until next week same time same place good luck good night guys