 Oh, forgot to plug in my sound, no wonder there's no beep. So today, in theory, should be technically reviewed, oh, the little static there made you jump? Okay. It says I just plugged in something into the audio jack, I realized I hadn't plugged in my speakers and I might need those. We're going to do a quick review of function notation. We teach you function notation in grade 10, and I have to be completely honest, it's stupid there, it's out of context. We kind of revisit it a little bit in math 11, and even there it's out of context. In this unit, you'll start to see why we like function notation, and those of you that are in calculus, which I think in this block is none of you, you'll find their function notation really sourced, because what this is going to let us do is generalize procedures for any function f of x, and then whatever graph or curve or shape I give you, you can kind of say, oh, I know the rules for anything. So very quickly, a quick review of function notation. We did some yesterday in our notes, but I thought I'd brush the cobwebs off. The first thing that we introduce function notation to you for in grade 10 is as a shortcut for substitution, and I guess in grade 10, that's kind of useful. So we give you if, and we pronounce this f of x, if f of x equals x squared plus 4x plus 5, find the following in simplest form, and then it says f of 2. What does f of 2 mean? Well, it means find function f, and wherever there used to be an x, replace every single x with a 2. We're going to write it out just this once, but we're going to try and, for the most part, do some of these in our head if we can. So what this really means is 2 squared plus 4 times 2 plus 5. And then in math 10, you happily evaluated it. 2 squared 4 plus 8, 12, 17. This early in the morning, I always wanted group consensus. But what we're also going to do this year, Amanda, is use function notation to generate new equations. You see where the 2 is this time? It is not inside the function. We're not replacing the x. It is in front of the function. This is saying what's 2 f of x? And it's actually fairly simple. What that really means is 2 times f of x, which was x squared plus 4x plus 5. That's what 2 f of x is. And, Nicole, I can simplify that by getting rid of brackets. It's actually 2x squared plus 8x plus 10. I would argue, though, Nicole, that's way shorter to write. And you've had me enough in physics to know if there's a shorter, cleaner, better way to write something. I'll usually choose or prefer that. What's the difference between b and c? Where is the 2 this time? It's inside the brackets. Now, there is still an x there. What this is saying is replace every single x with a 2x. And have you heard me say the word replace a few times? We used the replacement method last lesson for sliding, which is also why function notation works so well. What this really means is f of 2x is going to be 2x all squared plus 4 times 2x plus 5. That's f of 2x, except I can tidy this up a bit, Amanda. 2 squared is 4x squared is x squared plus 8x plus 5. That's the brand new equation that I get when I replace every x with a 2x. Also, replacing x with 2x changes your graph. Now, last day, we learned replacing x with, well, this one here, for example. If I look at E, that moves the graph to right. Next lesson, we're going to learn what this does, but I'll give you a hint. It's a horizontal compression factor of a half. It shrinks your graph this way. D, f of x plus 2 is the 2 inside the function inside the x? Say no. All we're doing here then is this, Andrew. Rewrite the original function f of x, which is 4x plus x squared plus 4x plus 5, and then plus 2 to it. Of course, I can say, Andrew, I can tidy this up a little bit. The 2 and the 5 are like terms. I would write this as x squared plus 4x plus 7. But for what it's worth, whatever this graph looks like, and Kirsten, I don't know I know it's a parabola, but I don't know exactly what it looks like. This graph got moved 2 up, which is why the function notation I think is way cleaner. I can't tell as easy that this is the equation moved 2 up, but I can sure tell it there at a glance. E, well, now we're replacing every single x with an x minus 2. This would look like this. x minus 2 squared plus 4 bracket x minus 2 plus 5. I can tidy this up a fair bit. x minus 2 squared, that's x minus 2 times x minus 2. If you foil that out and I'm running out of room, so I'm just going to foil it out, I think you get x squared minus 4x plus 4 if you go x minus 2 times x minus 2. And then you have another plus 4x from this term right here, and a minus 8 from this term right here, and a plus 5. This equation eventually simplifies Sabrina into x squared. I think the 4x is canceled, and I got a plus 4 minus 8 plus 5. I think I got a 9 minus 8 plus 1. And Steph, I'm going to argue I almost don't like this. This right away, I know it's been moved to right. If you gave me this, I wouldn't necessarily know right away that it had been moved to right. I'd have to somehow do a lot of algebra and do a lot of thinking to figure that out. And the last one, f of negative x. This says replace every single x with a negative x. Today you'll learn that that does something to a graph as well. It's a reflection. The negative is it next to the x or next to the y? It's meant to be a really obvious question. The negative is it next to the x or is it next to the y? Next to the x. As it turns out, and you did not do this with the parabola last year, replacing x with negative x does this. Clips it horizontally, 180 degrees. You know why you didn't do it with a parabola last year? When you flip a parabola, you get the same thing, so it's boring. But for graphs that don't have an axis of symmetry, it's much more interesting. Anyways, this would look like this. Replace the first x with a negative x, all squared, plus four times negative x, plus five. My new equation, now it's interesting, even though I've replaced the next with a negative x in the first term, what happens to the negative in this first term? It's going to vanish. I've done this question as a multiple choice question on test before. One of the first answers I have is negative x squared minus four x, plus five, and a lot of kids pick that because they, oh, the negative's got a little bit, no, no, the negative vanish is in the first term. Don't freak out, it happens. X squared minus four x, plus five, plus sign with a little bit more like a plus sign. All right, now let's do an actual equation again. Because f of x is equal to the absolute value of x, what did the absolute value graph look like? I can't remember. Let's write this in function notation. So wherever I see absolute value of x, I think f of x, I'll give you a hint. This first one I would write as follows, y equals f of x minus four. That's what we're doing. And Shannon, that moves the absolute value graph four down. Just like we're replacing x with what? If I wrote this in function notation, it would look like this. Y equals f of x minus four, where absolute value is my f of, moves it four right. What's my original function? Absolute value of x. There's a two there. Is the two inside the function? You would write this as y equals two f of x. Compare that with d. In d, the two is inside the function. If I wanted to write this in function notation, f of two x. By the end of the week, you'll know what each of these does to the graph. In fact, ideally you'll be able to glance at it and right away you'll say, oh, that's a vertical expansion by a factor of two. That's a horizontal compression by a factor of one half. Twice as tall, half as fat equals three minus f of x. I think absolute value is f of x and they got a three minus in front of it, so I'll put a three minus in front of it. What would that do, Mr. Dick? Vertical reflection and then three up, but that's later. Y equals, well, inside the function we have an x plus three. Outside the function we have a minus two. This one I do know. That's three left, two down. So three left, one, two, three, two, it would look something like that. Oh, this one. What was my original function? Absolute value, which is f of x, f of x, one over that. Y equals one over f of x. We're going to call that the reciprocal transformation. We'll look at that very, very last lesson of the unit. And h, I compare that to the original. It looks like they've replaced the x with what? I think negative x. So I think if I go to my function notation method I would go y equals f of negative x. So far so good. Happy campers all. Example three. A few tougher ones here, but you know what? It's very similar to what we did here, so I think you get any hang of it. We're actually going to turn the page. In math 11 you also talked about the inverse of a function. Now first of all we need to define the concept of inverse. An inverse operation undoes or is the opposite of an original operation. Stuff in English it sounds complicated. It's not. What's the inverse of plusing five? Minizing five. What's the inverse of timesing by six? Dividing by six. Ooh, what's the inverse of squaring? Square rooting. What's the inverse of sine? That second function inverse shift sign on your calculator has the inverse built in. Almost every single mathematical function out there has an opposite, an inverse. Then we also last year though started to talk about an inverse equation. An inverse equation was an equation that undid the previous one. An inverse equation, if you put five into the original equation and got 12 as an answer, if you put 12 into your inverse you should get five back as an answer. Five gives you 12, 12 gives you five. And there was a very, very easy way to find it. Does anybody remember last year how you could find the inverse? You're right, louder? Switch X and Y around. Switch the X and Y around. Or at the candy. Was there anybody else like oh the candy too? I went shopping and went to the doctor. Anyone else? Yes if you lied I would have asked you why and I would have remembered why so thank you. Now we're going to say that in math speak. You said switch the X and Y. An inverse function reverses this. The domain X of an inverse is the range Y's of the original. And the range of the inverse is the domain of the original. In fact we're going to use that trick to cut our memorization in half. And if they give us a yucky function and they say hey what's the domain? We'll actually graph the inverse and say oh the range is the new domain on the graph that I don't know how to do. Your range becomes your domain. Your domain becomes your range because you're switching the X and Y. If a function is defined by a set of ordered pairs you switch X and Y around. We're going to use that strategy quite often on a graph. If my graph goes through five comma two the inverse goes through two comma five. So if it's an equation it says interchange I just use the word switch. And in fact we can write it two ways. This is my preferred method. I like that because instead of Y equals F of X, Carson what do I have here? I've switched the X and Y around it's obvious. I wish that would become the universal method. It's clear, it's good notation. As soon as I see it it looks weird because the Y isn't where it normally is so right away it gets my attention and then oh yeah man I switched the X and Y around. Unfortunately the people that are in charge of math picked what I think is the stupidest possible symbol. Because the other symbol for inverse is this stupid thing. Y equals F little negative one of this. It's not an exponent. Looks like an exponent. It's not an exponent. But it's a number. It's a minus one. It's not an exponent. And honestly almost always to me math notation is clean and makes sense. I don't understand why they went with this. If nothing else why didn't they put a symbol there like an I for inverse or a Greek letter or something but the fact that there's a number there confuses people because this is one over X. This is inverse and it doesn't look that different and the fact that we're going to be doing one over X in this unit and we're going to be doing inverse I already know kids are going to get them mixed up. And unfortunately if you get them mixed up I have to give you zero on each question because you've done each question totally wrong even though you just got the concept mixed up but you knew how to do each concept it drives me crazy. So example four it says find the inverse of the following set of ordered pairs find the inverse of this. How do I find an inverse Andrew? So instead of negative four negative two it's going to be negative two negative four that's the inverse of that first point. Make sure you don't change positive negative it's just literally switch the numbers around that one doesn't become positive negative they keep their sign negative two negative one is going to be negative one negative two. Negative one zero is going to be zero negative one zero comma one is going to become one comma zero two comma four is going to become four comma two three comma eight is going to become eight comma three then it says join each set of ordered pairs with a smooth curve. Okay I've seen that graph before it's the wounded seagull oh no no it's the square is a square root graph of some type. Now if you can change colors we're going to sketch the original points as well just so you can see was that a phone where was that that was yours raise your hand if you want 10 bits on Friday. Thank you sir that was you know what that was so gracious of you to take one for the team to put the team in front of your own selfish interests very very selfless of you you know what you'll have more friends that well you'll have a friend that way start out simple you know maybe if you have a different color I have a different color I'm going to go with blue I'm going to graph the original points here now original points are negative four negative two negative two negative one negative one zero zero comma one two comma four and three comma eight the original graph looks like this and you can see I hope I hope I hope that the inverse is symmetrical with the original in fact we say it reflects about a line there is a line of reflection and the line of reflection is this one right there if I could draw a little better and the reason I'm showing that to you is it's a handy dandy built-in error check Amanda if your inverse is not a nice reflection of the original you must point out what's the equation of this line here well this line goes through one one two two three three fact this is the equation y equals x the inverse is a reflection about the line y equals x which step kind of makes sense to me because Andrew how do I find the inverse what are you now I'm going to say that must be let the y be x and the x be why that you know what doesn't surprise that that line shows up on the graph as a reflection line because we actually did that operation we switch the x and y let the y be x and the x be y I will be asking to graph an inverse I will also ask you to find an inverse equation so example 5a says find an inverse this is the only time that I think function notation is more cumbersome is tougher because what letter is sitting right there what letter is sitting right there I worry I'm going to get them confused so this is the only time where I ditched the function notation and put the y there instead why Trevor Trevor how do I find an inverse switch I got a y now if I didn't have that I do the inverse equation is going to be x equals 3 y plus 2 but then we're going to almost always ask you to get the y by itself how would I get the y by itself here minus 2 from both sides and then divide by 3 I think y is going to be x minus 2 all divided by 3 or Kirsten they might write it as x over 3 minus 2 over I don't know they might split it up they might not I wouldn't if I was handwriting because why would I write more work than I need to but how can you check to see if you're right stick in a number if I put a 5 right there what's 3 times 5 15 plus 2 5 gives me a 17 if I put a 17 right here 17 minus 2 is 15 divided by 3 5 17 gives me 5 they're inverses put an x get a y put a y into your inverse get the f once again they want an inverse says find inverse find inverse I'll replace the f of x with a y okay how do I find an inverse switch okay so the first step I would do is I would go x equals y squared minus 1 then get the y by itself this one little tougher I'm probably not going to do this whole thing in my head I would plus 1 to both sides certainly what's the inverse of squaring okay y equals big square root of x plus 1 does anybody remember from math 11 it's obscure why I'm wrong Andrew what when you square root both sides of an algebraic equation you have to remember that you could have had a positive or a negative answer because it could have been a negative squared or positive we don't know the pause here temporarily I'm probably going to this cluster one does this cluster go to 950 I'm gonna give you about 20 minutes to work on this and we're gonna pick up with a bit more anyways I think you can try one six yeah one three five six you know what that won't take you 20 minutes only take you about 10 or 15