A LEVEL MATHS: Functions Exam Practice LESSON 26

hello welcome back to another one of the a level maths live stream so we're finally starting the A2 content and we're going to start off with functions which is a huge part of the a-level specification and it's going to bring together the exponentials logarithms and graphs Transformations and everything all into one that nice little package and it does kind of follow on from where it is in GCSE but of course you have the more advanced functions and you also have to talk more about the domain and ranges of functions which is slightly more tricky but with some examples that we're going to do we'll get right through it so first of all it says the function is f and J defined by f of x is 3x plus 4 and G of X is 2 over X plus three so the first thing we should look at is if you notice and you would have seen this on your GCSE exams too but you might not have known what it means this right here is something called The Domain and the domain is just all of the X values that we can have so all of the possible X which also you can call it as the input values right so what that means is what values can I put into this function right that are legitimate they don't give me an invalid results and if you notice it has this symbol here and then R so what this actually means this part here is it's the set of all real numbers and if you remember a real number is literally just any number from minus infinity to plus infinity it can be an integer it can be a decimal it can be anything okay uh technically I should have an s on the end okay so that should make sense right if I look at what x here is there any value of x that would give me a naughty number right something that would be unsolvable math error no right like I can put in negative infinity plus infinity in every single number in between and get a valid output value so the range of values is infinite as well okay and if you're looking for G of X we have X is any real number it's the set of all real numbers except three and if you look here if I subbed in 3 here technically this would actually be minus three for this question but if I sub in minus 3 here I would get minus 3 plus 3 is 0 2 divided by zero is a math error so that is why we get these prompt a it says evaluate f g of one so again there's a bit of a recap from GCSE but essentially you work from the inside out so if you work out what G of 1 is first so that's just two over one plus three so two over one plus three is two of four which is one over two okay and then all you need to do is sub whatever that value is back into f of x so three times a half plus four which is three over two plus four that'd be eight over two so it should be eleven over two overall okay nice and easy Part B then says solve the equation of g f of x is equal to six so this is slightly more involved what we need to do is work out what G the equation is with g f of x so what this means is I take all of the function for f the entire thing and I substitute it in place of X in this expression here okay so if I write that out now we have two over and instead of X I'm now writing 3x Plus 4. and then plus three again on the end which is of course equal to 2 over 3x plus 7. and now in the question it says we need to solve for when this whole expression equals six so I'm going to equate that to six and now it's just a simple linear equation that we need to solve okay so again when you have g f of x you're substituting in the second function into the outer function first function okay so I'm going to times both sides by 3x Plus 7. and we have 2 is equal to six three X plus seven I'm going to take it nice and slow just to make sure we don't make any mistakes 6 times 3 is 18 add the X on 6 times 7 is 44. sorry 42 not 44 what am I talking about 42 and subtract 42 from both sides so let's write that out we get negative 40 equals 18x so X is just equal to 40 over 18. and of course that's a negative so negative 40 over 18 so you can keep halfing it if you wanted to or you can use a calculator to work it out because a level exams are all calculator so let me just open up the calculator so we ready to negative 40 over 18. again minus 20 over nine so yeah you can either half it or you can use a calculator so minus 20 over 9. perfect and that'll be it so six marks It's relatively quick not too bad I like to think anyway question two if you have a look the function f is defined by this quadratic and once again we have X is a set of any real number but it has to be greater than or equal to negative one now sometimes you might look at this and think why is that the case um sometimes they just place the restrictions themselves so they'll restrict what values you can have for X intentionally but we have a new word here the range so what is the range well the range is all of the output values but values so if I put in any of these domain numbers what is this range of values I can get out of the end right so if I sub in -1 then 0 and then one Etc what will the set of all the values now with the quadratic it's actually very difficult to see what that is just by looking at it right because you're saying X has to be greater than or equal to negative one so you can probably work it out by just subbing in negative one and then working out from there but that doesn't work always so I'm going to show you the way on how it works always so again in this case they make it slightly Easier by showing you this okay however if you imagine that isn't there here's what you would do you would do completing the square or factorization in order to work it out so you'd always do factorization sorry completing the square um and because the reason for this is that completing the square gives you the minimum point of a function a quadratic function so that would be X Plus 1 squared minus 1 squared and then plus one on the end so we just get X Plus 1 squared so my minimum point is at -1 0. so what are the range of values that I can get well this is the smallest y value right that's the smallest output value I can get which means f of x has to always be greater than or equal to zero because it can only get bigger because the smallest value is zero so again you could have seen that just by subbing this into this equation here however you don't always get given that okay sometimes I'll just give you a quadratic and they won't give you this part and you need to State the range best thing to do is just find the minimum Point how do you find the minimum point you do completing the square and you may notice that in fact in this question specifically you could have just done regular factorization but yeah Part B it says you need to sketch the graphs of f of x and F minus one of X on the same diagram so let's have a look see so I'm going to sketch f of x first which is X Plus 1 squared and it crosses the x-axis just once at minus one so you'd have a graph that looks I mean to be fair I'll probably just want to do something that goes like this okay and it should also cross y so if I sub in x equals zero to get the y-intercept across there at one now what is the inverse function and what does it look like on a diagram well the inverse function is the function that will give us the exact opposite of this so in this question here in this initial equation I put in X values and I get out y values the inverse function is if I put in the Y values I should be able to get the X values out so it's the function that does the exact opposite of an initial function and it turns out that that is just a reflection so F minus 1 is just a reflection in the Y equals X line so this means the coordinate swap place so what I mean by that is and it's not place and sign so what I mean by that for example here the x coordinate is so we have minus one zero so now we swap those around we get 0 minus one because all I've done is I've taken the values and I've swapped them around here we have zero one so what we're going to have is one zero okay now if we actually look at this as being a reflection it should look something like that so again all you're doing is reflecting the shape in the Y equals X line so you draw the exact kind of opposite if you can but the way you can help yourself and keep in mind that in the exam you will have to actually um do this yourself as well is you need to actually swap the coordinate places so again Zero moves to where the minus one is the minus one moves to the zero to make this okay so you're basically just swapping it around okay and you do that for every set of coordinates so the minimum point the X and Y intercepts okay perfect well technically there's only ever one y intercept but we'll go into that bit later part C it says find an expression for f minus one of X and State's domain so here's what I do first of all I write the equation out again but instead of f of x I'm going to write y now the whole goal here is to rearrange for x okay so I need to get x equals something so the first thing we're going to do is I'm going to use that completing the square form because that's really convenient for me right X Plus 1 squared and that is the same thing so if I want to rearrange for X I'm going to now square root both sides so now I get square root y equals X plus one minus one from both sides and I now have X is equal to square root y minus 1. and then I replace X with F minus 1 of x and Y with X okay so this is actually relatively simple um equation they've given me but it's still four marks so it's pretty beefy um but it can be kind of tricky to rearrange these because rearranging quadratics isn't something you may have done before because it's a bit of an obscure thing but as you practice you'll get kind of better at this but you have to do things like logarithms and exponentials so my inverse function looks exactly like I forgot about that x minus one and it asks me what's the domain so what values can I have well again what you could do is say well logically speaking I can't square a negative number so X must be greater than equal to zero and you would be right but give me a second that isn't good enough okay because it may work in some situations but it doesn't work in all situations so what I prefer to do is use that logic I just did to check my answer so if I get an answer and I apply the logic and I think okay does that make sense and it does then I'm good if it doesn't make sense then I need to have a look again on my method here's what you need to know right if you have a domain and range of the original function they swap places for inverse functions so for the inverse function the domain equals the range of function so of the original function so if I have a look back at my range here I said f of x is greater than equal to zero the domain of the inverse is greater than equal to zero it's the same thing and likewise the range is just the domain of the function so what would the range be of this well the range would be any real number but it has to be greater than equal to negative one so I could just write that as f of x is greater than equal to negative one okay so hopefully that makes sense but again the domain and range of swap places and that should hopefully make sense right because I said that the original function takes X and gives you y the inverse function takes Y and gives you X so those should swap around okay so let's go to the next one the function f is defined by this expression where X greater than 0.5 hopefully this makes sense because you can't take the log of a negative number and you also can't take the log of zero so that should hopefully make sense part A find the value of f f of one so I would find F of one initially so that is and it says the exact value because it's a logarithm right so we have 2 1 minus one so that just gives me two Ln whoops Plus Ln one okay now ln1 you should hopefully know is just equal to zero so this is equals two and then I'm going to do the same thing again so I'm going to take this result and plug it back into the original expression and that will give me 2 plus Ln 2 times 2 minus one so that gives me two plus Ln three which is my final answer for two marks so again these function questions they generally ask you just as a substitution and they're generally going to ask you about the domain and range and inverse functions uh so any one of those combinations so if you can do that for as many functions as humanly possible so what I'd actually recommend is even for the questions that don't give us domain and range as a question just as practice for your vision just write it out anyway so finding the expression for the inverse function again my method is first to just rewrite the original but with Y instead of f of x and now I'm going to try and rearrange 4 x so with logarithms it's actually not as bad as it may look what you want to do is get rid of everything next to the logarithm so if it's multiplying I think subtracting whatever so I'm going to minus 2 from both sides and we get y minus 2 equals Ln 2x minus 1. the next thing you want to do is something called exponentiating so you're raising everything up to the power of e so we have e to the Y minus 2 and the cool thing about that is it actually just gets rid of the natural logarithm so we're just left with 2x minus 1. so now we can just keep moving over and we get uh so plus one so we get e to the Y minus two plus one equals 2x so then X is just equal to and you can write this in kind of any way you'd like but this is probably the best way to do it like that so my inverse function now is just F minus 1 of x equals e and this is where you need to be really careful is you need to replace the Y with X as well so some people leave it as Y which does deduction marks unfortunately but that would be your final answer so again not too shabby I don't think three marks nice and quick I'd say perfect so for the next one the functions f and g are defined by f of x is just e to the x nice and X can be any real number which makes sense right you can put any number in for em the power of E and G of X is 2x plus Ln x x has to be positive y well again you can't take the log of a of zero or a negative number so write down the range of f so if I think this is just the e to the X graph are there any values that it can't take well a good thing to do is to sketch it because I mean and this is literally why people sketch graphs in the first place because it helps you visualize exactly what's happening if I sketch the etx graph I get something along the lines of this and this it shoots off to Infinity but if I notice it will get infinitely close to the x-axis but never below it because this is asymptotic so what am I trying to say well it's always greater than zero and that's exactly what the range is f of x is greater than zero because again if I were to draw the graph the line or the curve in this case never crosses or even touches the x-axis so it means the Y value always has to be at least a bit positive perfect for Part B it says find an expression for the composite function GF of X so what does that mean well we're subbing all of f of x into G of X so let's do that so g f of x is going to be so instead of X I'm going to write ethx Ln e to the X and what happens when you take the natural log of an exponential it gets rid of it so these two things basically cancel each other out but you still get left with the X you have 2 e to the X Plus X for that question and then it says write down the range of GF of X well this is where things get a bit more interesting if I think about this value this kind of uh graph here it's pretty hard to draw I will say but if you notice as this gets in this way you need to start using your logic right so as X increases right GF of X is just going to keep increasing forever right because they kind of agree with each other if you'd like and it's going to exponentially get higher so we know that g f of x can be positive can it be zero yes it can why because this x can be negative right now this can only ever be positive but it gets smaller and smaller and smaller and smaller as you put negative values in whereas this is just going to stay getting bigger and bigger and bigger when you put negative values so if I put in minus infinity for example this thing gets really close to zero and this thing gets negatively infinitely big so essentially GF of X can take any number so how do you write that well all you do is you put this e-looking thing and you say that just means it's the set of any real numbers and the r you generally like make it look a bit fancy because it's not you know just R the other way you can write it is some people prefer to do a double r like this and that also works as well okay I hope that makes sense so for this one the function f of x is defined by sorry function f is defined by one over X plus two and of course X can't equal negative 2 because otherwise naughty things happen we get a math error write down the range of f of x so this shouldn't be too bad first of all if I make X really close to zero it can just be kind of a half if I make it negative it can be any negative number if I make it positive it can be any positive number so basically and I can also make it really really small so f of x can be any real number however there is one number it can't be ever think about it when you're dividing numbers what's the only number you can never actually get as an answer well you can never ever get zero you can't get zero by dividing it's impossible so that is the only value that f of x cannot take cool so that was I think quite nice and easy now if you'd like if you look here finding expression for f of minus one of X and State's domain well it's domain immediately is just going to be X doesn't equal to zero but X can be any real number that's the domain right why because the domain of the inverse is just the range of the regular and vice versa so the domain of the inverse is just the range of the original value here so I can write that straight away get a Mark and I haven't done anything yet but let's uh get down to business so we have y is equal to 1 over X plus two and we need to rearrange so what might be better is I mean this actually is quite straightforward times both sides by X plus two so we get y X plus two is one so what I could quite easily do here is just divide both sides by y so I get X plus two equals 1 over y X is equal to 1 over y minus 2. and you could make this a bit more interesting by combining the two fractions so if I were combining the two fractions you would get something along the lines of um one minus two y over y so that also works as well but either one of these is correct so then my actual answer is f minus 1 of X is equal to and again either one that you want to do so I like to compare so I combine fractions but yeah it just says find an expression so you can use either one of these so I'm going to write or equals um one over x minus two both those are the exact same thing okay sounds nice and easy and we've done the domain up here already perfection plus it says uh sorry we have G of X is equal to x squared minus five solve f g of X is equal to one half so the first thing to do is you're taking G and you're subbing it into X up here so we have 1 over x squared minus 5 that is the G part and then add on 2. so that equals 1 over x squared minus three and what does that equal to equals to a half okay interesting so we're more than likely going to get some kind of quadratic from this so I mean there is actually one thing we can do is we can cross multiply so what I mean by that is when the times both sides by 2 to bring the two to the top so actually I'll do it in two steps just to make it a bit more simple but I'm going to show you quickly and he turns both sides about x squared minus 3 unit 2 is equal to So eventually once you're happy you can jump straight to this line and okay let's have a look so we're going to add three to both sides we get 5 is equal to x squared so then X is just equal to the square root of five but keep in mind that is a plus or minus value because when a level maths now we need to make sure we're taking into account every possibility so either plus or minus which makes sense there should always be two solutions for a quadratic unless they're the same okie dokie one more so we need to find the range of this and if you notice unlike in the first question they gave me a domain which is any real number so we have to do completing the square so like I said earlier sometimes they're going to force your hand and they're going to force you to do the actual um completing the square method so f of x is equal to and if you remember to do completing the square the formula is like this so we have x squared plus BX plus C then what you would have is X plus b over 2 squared minus B over 2 squared plus C okay so we have X plus two squared minus 2 squared plus one so if I were to simplify this God I wish I could talk squared minus four plus one so we have X plus two squared minus three so what's the minimum Point here well the minimum all you do is you make the x value negative so the value inside this bracket make it make it the opposite sign and you just keep the other sign the same so the smallest y value is negative 3 which means f of x by definition has to be greater than or equal to minus three that's it so for the next part this is something that is extremely confusing for certain pupils and it's explained why the function f x doesn't have an inverse now there are two methods for what you want to do here one is to just remember this one sentence and you can use it every single time this question comes up and you're guaranteed to get a mark okay and it's this because it is a many to one function that will give you the mark in every single instance of this okay so if you uh so if you're watching this back on YouTube a bit later uh skip ahead to whatever time stamp I've put on screen now if you don't care right if you you're just here to get you know Smash and grab get the grade then this is the sentence you need but if you want the kind of reasoning behind this I'm gonna go through it now so again you feel free to skip ahead um if you're watching the live stream live then you can't skip ahead you're on this race with me so many to one functions what that means is is many inputs give one output which if you think about a quadratic makes a lot of sense right so if I were to draw this out we have a function that looks something vaguely similar to this where this here is -2 minus three and as you may be aware this is a symmetrical function right so I equal points on either side of the graph have the same y value so for example this could be y equals one they're both equal to one here it could be three and this could be three right the point is for any one y value there's two x values and this should also make sense based on what we had earlier so when we did the solution here we got plus or minus value right so for one y value there are two x values that give me that value obviously the exception here is the minimum point right but the point is is if I am working out an x value I'm going to take a square root at some point which gives me plus or minus values which should make sense because it's completely symmetrical now I told you before that if we then take the inverse this thing swaps so what that means is it becomes a 1 to many function which is impossible so what this means is a single input gives many different outputs so try and imagine what that will look like well it means for the same x value you can get any y value so again if I were to draw that I could put in x equals 1 right and I could get this value and this value okay and essentially what you'll be getting is you'll be getting kind of like a loop like this which doesn't actually appear in real life and the reason for that is because think about it imagine if you did a calculation on your calculator right like for example when I go on a calculator I know that if I type in 3 times 6 I get 18. and no matter how many times I press equals I get 18. with a one to many function I could put in something like this I could put in a set of numbers and every time I press equals I'll get a different answer hopefully now you can see why that's a bit weird because I can do the same mathematical calculation and get different answers every time so it's pointless right doesn't it just doesn't make any sense so hopefully that makes a bit of sense but now you're probably wondering if I go back to one of these early questions I drew what the function is so the thing is in your exam like here it can ask you to find inverse functions right and you can even ask you to sketch inverse functions but they don't exist right now imagine like again this is actually a really really good example right scroll x minus one well if I take the square root of 4 I get plus or minus 2. so if I do plus or minus two minus one I'm getting two completely different answers but how do I know which one to take it's impossible to distinguish right I'd have to make a conscious decision on which one to take do I take the positive value negative value so it doesn't really work as a mathematical function you need to have a human being make a logical Choice which you know isn't exactly the thing you want to do all the time like if you're programming a computer for example how's the computer going to decide right you have to put human logic into it I'm hoping that makes a bit of sense so it's annoying to me because on the spec it says teach them that it doesn't exist but then it also says you need to sketch them and be able to find them even though they don't exist but the point they're trying to make is they logically don't make sense and they logically don't exist so just as a note to myself I've stopped the rant about inverse functions of one-to-many uh sorry many to one functions so now we can keep going with what we're doing here so hopefully you're back we're going to go straight on to the next one sketch the graph of y equals to x minus three if you notice there's no question number why because this is also from a past paper this is from the 2021 paper I believe so sketch graph of y equals the magnitude of 2x minus three so this is another new thing that they add onto the specification for a level so what does this modulus mean what it means is make positive now this doesn't mean swap signs right so basically the modulus of four is four the modulus of negative four is four okay anything that's negative you make positive if it's already positive you do nothing about it so what does this look like in an actual graph well if I were to sketch the graph of 2x minus 3 okay I would get something along the lines of a value that is going to cross up minus three is going to have quite a steep oh God quite a steep curve like that uh line right and I'd want to work out what this x-intercept is how do I work out the x intercept I say y equals zero so zero is equal to 2x minus three so 3 is equal to 2x so X is equal to three over two so this x coordinate here is three over two that's it right Happy Days now let's think about what this modular sign is doing right it's saying that I'm making all of this positive so my y values all become positive so if I have a look here Y is negative underneath the x-axis so all of that is magically made positives instead of -3 I have Plus 3. right and all of these coordinates will be made positive so if I match them up that one this one here will come up here this one here up here so what it actually looks like is that okay so what you're going to do is basically get rid of the actual solution of that it should look a bit more even than that but I'm terrible at drawing as you are more than well aware by now that's all it looks like it effectively it's not a reflection so the way that some people explain it is it's a reflection in the x-axis it's not because this bit stays positive it's more like it's bouncing off so if any of you do physics it's like you're drawing a ray diagram for a reflection right so literally it just bounces off so imagine it literally hitting bouncing off somewhere okay or you can be a sane person and just say okay all the coordinates are positive for y so make all y values make only the Y coordinates right we still have negative x coordinates to make Y coordinates positive okay and it's just like it's bouncing off okay my voice is killing me right now so what does it mean for actually solving equations though well this is where it gets a bit funky what you would do here right is you should accept that there's two solutions there's one where it's face value right there's one where it's 2x minus three is less than nine however because I'm making this whole thing positive regardless it's equally possible for me to have a negative solution it's just that the solution was made positive right so the solution is I mean the actual answer is positive but I could have a negative solution that is made positive it's forced to be positive so all you do is you stick a negative in front that's it you're gonna have always have double the solutions so if it's a linear you're going to have two if the quadratic you're gonna have four okay so solving this should be easy as Sin right now so 2x less than 12 x is less than six you're also going to have the exact opposite so negative 2x plus 3 less than nine so just subtract 3 so you have negative 2x is less than six now X is greater than minus three what happened there when you remember this because this is something Ayla was doing to forget because you get toward it in about year nine or year eight when you divide or multiply by a negative you flip the inequality put the direction or how about this how do we really say change instead let's make it a bit a bit nicer you change the direction of the inequality okay just the direction right so if this was less than equals to this would be this would still be equals to it just changes which way around your facing okay if you don't like that you could add 2x to both sides and subtract six you get the same answer but you know why make things longer right okay so that's how you would do it for something like that and again it's the same thing if it has x's on both sides so again you're going to have 2X minus three is less than X plus one whoops and you're also going to have this same thing here and it's again like I said identical it's completely identical right so solve this you get so move this x to the other side we get x minus three but there's some one so X is less than four here move this 2x to the other side move the one to the other side we get 2 is less than 3x so that means X is greater than two thirds nice and easy two marks very Snappy very quick okay so I hope that's starting to make a bit more sense so realistically the the main challenging ones to rearrange your inverse are the inner quadratics and then logarithms and exponentials but once you've done it a bit it's quite nice you know it's not too bad so for now what we're going to do is leave it and I'll see you in the next one for sure so