Greetings, my dear friend, my 9th-grade class! In this video, we'll be reviewing functions from beginning to end. This will be something different from the usual full reviews; I want it to be more like a private lesson. We'll start with a blank sheet of paper. Imagine this: You say, "Teacher, I have an exam the next day, could you remind me of the important points about functions, touch on the key points, and review the areas where questions might come up?" You sit next to me, and I say, "My dear friend, come on, you liked the RM group, right? You commented on the video, so let's sit down and review functions together the night before the exam." Of course, I've made about 15-20 videos on functions, and I absolutely won't skip to this video without watching them. But still, if you say, "Teacher, what did we do last night? We made a mistake, we didn't watch those, I'll watch this one, I accept it." At least I'll explain everything about functions to you. " Teacher, I've watched all of them! You're great! Let's quickly review them together then!" I said, " I'll quickly review functions, I'll only emphasize the points that the Ministry of Education textbook requires from us." Look, you'll remember from the Ministry of Education's textbook, there are checkpoints. First, let me show you where I'm going to explain this. F is a linear reference function from real numbers to real numbers, FX is a function given as x. We call this a linear reference function. The textbook says to explain everything a student needs to know about this. Do you know where I'm going to explain this first? Let me explain it like a question. I said question, so we're starting with functions. F is a linear reference function from real numbers to real numbers. I have the function FX = x. Let's open a parenthesis here. Interestingly, the map is trying to explain it without going into the definition of the function. Starting from first-degree equations in 8th grade, assuming that y is on the other side of the equation, it says we're explaining this as a function. Actually, it's not like that. The real definition is this: there's a domain and an image set, sets a and b. The elements a and b taken from A must be non-empty. Of course, every element taken from A matches only one element in B. We need a relation that satisfies this, but we do n't know that relation because it doesn't exist. It's not in the curriculum, we're just going around it now. The definition is called a function from A to B. Now, let's talk about the function from A to B part. Don't worry about that. What does the Ministry of Education want from us in the curriculum? From real numbers to real numbers. We defined FX = x, right? Here, R will be my domain, that is, where I will choose x. Here, R will be the image set, we will find its images. Thus, we will find the images of that X. So, what did you say, teacher? In essence, we took x, transferred it to the image set, and matched it. Let's try to see the logic of this together. First, I started with x as 0. If I give x 0, and write 0 wherever I see x in the F function, the answer of F 0 is 0. Actually, do you know what I found? I found the answer to y. This expression is also shown as y = FX, and then we write this part, which we will call the rule of the function. Actually, y's are equal to x's, or x + 1's, or 2x + 3's. If I had to write the rule, okay? Now, X's... Let's increase its value. What happens if I give x the value of 1? Sir, when I give x the value of 1, the answer of f1 is 1 and the answer of y is 1. Essentially, my function always consists of ordered pairs. From the ordered pairs X and Y, I chose x as 0, y as 0, I chose x as 1, y as 1. Great! If I choose x as 2, what will the value of the function be? It will be 2. So my function will pass through the point 2 to 2. Okay, we always gave positive values ​​to x. Let's give a negative one too. If I write -1 where I see x in f, the value of the function, which is also the value of the function, will be -1. Notice that -1 passes through -1. If I write 2 instead of x, the value of f-2 becomes -2, so our y value becomes -2. So it passes through the point -2 to -2. Then the function manager wants this from us. My friend says, let's put this in a table. Look, it's very simple. Don't scare yourself. Think of it as a number line. Essentially, it's a number line where the values ​​related to x are placed. I'm going to write the results of the FX function here. Now, when I give x 0, 1, 2, or -1, I find the values. Besides that, and I know this number line is positive infinity in this direction and negative infinity in that direction, right? Now, when I give x -2, what's the answer in the F function? -2. When I give x -1, it's -1. When I give x 0, it's 0. When I give x 1, it's 1. When I give x 2, what's the answer? 2. Actually, Map is subtly showing you a sign table here. When I say x = 0, it says, "Let's draw another line from here. When we give x 0, what's the value of the function? 0. Let's write 0 on that line." It really continues like that. Then it says, "Pay attention to the values ​​here. What are these values? They're all positive. So if I give values ​​greater than 0, they're positive, and if I give values ​​less than 0, they're negative." There you go, the sign table of the function. Look, subtly, what information have I given you here? I've given the sign table of the function, and also the value table above. We can draw the graph of the function using the value table or sign table. Now it's time to draw the graph of the function, and especially the map requires me to know its graph very, very well. So what is its graph? Look, I've drawn the x-axis, I've drawn the y-axis. Where do I need to pass through? Look, it says here, I've marked it from 0 to 0, I need to pass through the 1 to 1 point, I've marked it. I need to pass through the 2 to 2 point, I've marked that too. Similarly, we need to pass through the -1 to -1 point. After marking that and drawing the line, if we draw this line, for example, is there a value between 1 and 2 that passes through 1.5? Absolutely. How do you know that, sir? Look, write all the real numbers instead of x. We don't have enough time to write them all down. We say, let's write a few of them, connect them and draw the line. What did we do? We drew the line, that's it. Now you 'll write the function on top of it in the exam. Look, the graph of FX = x is one thing, that's it. Sometimes it's also written as y = vx. Did we understand each other? Any problems here? Great. Now, let's give some information for the same function or before. For a function, I've put the x-axis here. Beyond the x-axis, draw the graph of the function like this. Look, any function. You don't know which function it is, I only know it's the FX function. There's a point where this FX function intersects the x-axis, okay? After it intersects the x-axis, if my function is in a region above the x-axis, the values ​​of the function here are positive. We represent the values ​​of the function with FX. Notice that the values ​​of FX here are greater than 0. If the graph of my function continues below the x-axis, just like here, the values ​​of my function will always be negative. How do we show this? The FX function... I couldn't write it, one second, I'll write it. Hopefully, the values ​​of the FX function here are less than 0. Sir, right at this point, where it intersects the x-axis, the value of FX is 0. How do I know this? Look, I showed it here too. Did you see? Here, the function's value is 0, where x = 0. I don't know which function this is, so I won't know where it intersects. Look, what we did was, we called the intersection point 0. Now, let's go back to the function itself. Where are the values ​​of this function? On the x-axis, when x is greater than 0, what values ​​does our function take in this range? Values ​​greater than 0. What values ​​does our function take here? Values less than 0. What is the value of my function at this exact point? It's 0. Look, I'll say it again, if the word "value" comes out of my mouth, I'm actually always talking about y. It's the FX function, specifically the linear reference function F that it emphasizes. So, besides that, what else do I need to know? I need to know the qualitative properties of the function. Copy this from here, let's try to write down the qualitative properties of this function together. Now, paste this together: the domain of the function, but which domain, sir? The widest domain. What does the widest domain mean? Where does the graph of this function start at the bottom, at x? Find that for me. Domain means look at the x's. Look, let's write it in parentheses next to it. Look, here's x... It means he's asking me for it. Agreed? The domain is asking me for x, sir. Look at my graph, there are x values, x values, x values ​​opposite it. Do you see the x values ​​here? Okay, how many are there? You don't know. Actually, how far does this graph go? To infinity. So where do I start with the x values? I start from negative infinity. Okay, how far do I go? Look, I'm drawing perpendiculars from the graph to the x-axis. The graph goes to infinity here. And where do I go with the x values? Notice, I go to positive infinity. If you want, I can write it like this, sir. Let's write our domain. The domain can be negative infinity and positive infinity, but we don't write it like that. The place we call negative infinity and positive infinity is all the real numbers, right? Exactly. The widest domain of my function is all the real numbers. Okay, can you write the range of my function, sir? Again, when the range is mentioned, only one thing will come to your mind, just the y-axis. So I'm looking at the y-axis in the graph. Look, I'm drawing perpendiculars from the graph to the y-axis. Where did I start? From the very bottom, negative infinity. But sir, this isn't negative infinity, it goes to infinity, okay? The graph goes on and on. Some professors even say, "If you don't add an arrow to the end of it, I won't accept it. Please, add the arrow, okay?" There are professors who insist on it. Now, I'm drawing perpendiculars from here, drawing perpendiculars... Where does this go? To plus infinity. So, what can be written in the image set? Between minus infinity and plus infinity. In short, it can be said that it's all the real numbers. Any objections up to this point? Okay, so the function's II means the place where the function's value is 0. It asks you when FX is equal to 0. And our function is the function. Where did we get 0? Look, we got it when we gave x 0. So, where is the function's 0? It's written like this: x = 0, okay? Our function's 0 is x = 0, or you can write f0 = 0. The first one above... I say first one, the sign of the specific function. Actually, the sign of the function means the sign table here, that is, where the function takes positive values ​​and where the function takes negative values. Aa Really, pay attention to how my function takes values ​​on the right side of 0. Positive values. And here we 've shown that it really takes positive values. At x = 0, the function's value is 0. Where is the part below the x-axis, sir? Where x takes negative values, look, this symbolizes x. Where x is negative, this symbolizes y. What happened to my y values? Negative. Really, if I'm below the x-axis, those areas will always take negative values. Okay, we've come to the maximum point of the function. Here I'm going to do something like this: I need an extra page. There should be something called "add page". Here... Wait a second, was it added from here? We're confused. Let's add a page here. Adding a page would be good, but it's not enough, I'll explain it to you in detail, don't worry. We'll review the whole thing. I'm continuing from below. What am I looking for in this function? Where was that function's maximum point? I'm looking for the maximum point of the function. If you want to find the maximum point of a function, you just need to look at its graph, and we'll always be working with graphs anyway. Being able to draw it from memory. By the way, being able to draw the function graph I just drew from memory. Look... So, what is the graph of y = FX = x? Look, here it's doing an angle bisector, dividing 45 degrees into 45 degrees. This means the maximum value point of the function, where is the greatest value it takes in x? That's what it's trying to ask me. But it doesn't exist, sir, it goes on infinitely, so you 'll write that this function has no maximum point. Agreed? Okay, sir, does this function have a minimum point? It's asking where this function goes at its lowest point. Is there such a place? No. So, what is the minimum point of this function? Again, there isn't, but are there situations where there is? I'll give you a very nice example, don't rush. Now, let's look at the one-to-one relationship of the function. One-to-one is the normal condition. When you say one-to-one relationship of a function, mathematically it's defined as follows: F is a function defined from A to B. For every x1 and x2 element of a, I will choose two x values ​​from the element of A. If the values ​​of x1 and x2 are different, then fx1 must be different from fx2. In short, this means that for every different x in the function, you must find a different y value. Do you really have an x ​​value where you find the same y value here? For example, if you give x -1 and -2, does x also come out as -1? No, there's no such thing. So how will I understand this, sir? We will understand it from the graph, and I will call it the horizontal line test. How will we check for one-to-one correspondence? Horizontal line test. Now, this is the best way to understand it. What do I have of the function? I have its graph. Look, I'm drawing the graph again. It doesn't reach here at all. I'm drawing these, the graph of y = x. Is that right? Also, y = FX = x. By horizontal line, I mean I'm cutting the graph with lines perpendicular to the y-axis. You can cut it from anywhere you want, you can draw as many lines as you want. How many points do you cut it at each time? It cuts at one point each time. So if it doesn't cut at more than one point, what will we call this function? We will say it's a one-to-one function. Agreed? Really, this function is a one-to-one function. What did I do? I sent my horizontal lines. What did we do each time? We cut at a single point. If we cut at a single point, my function is a one-to-one function. I will call this information one-to-one here. Below, I'll give you the information, what else do you want? It says find the intervals where it's increasing or decreasing. We'll decide by looking at the graph. Look at it again, the function, again, I draw it every time. Because these are important for us. I brought the function here, we drew FX, almost. It would be great if you took notes along with me. We will examine the graph of the function FX = x. Look, where is the graph of the function constantly going? It's going up, up, up, up. Does it deviate at all? Or does it deviate? It hasn't deviated, right? Then do you know what we call this function? This function is an increasing function, this function is not a decreasing function, it is an increasing function. Why? Because my graph, my friend, you're great. Okay, teacher, when does our function become decreasing? I will give an example of that in a moment, but let me also give you some mathematical data about increasing or decreasing functions. Let's take each x1 and x2 from the set a, that is, from the domain or a certain interval. If, as the x values ​​we take increase, the value of the function also increases, this function is called an increasing function. Some schools also give these definitions, friends, these are very valuable definitions. In mathematics, these are our essential examples. Look, I took the value of X1 from here, and the value of X2 from here. Notice that fx1 is here and fx2 is here, right? Actually, these are y- values, but it's the same thing. What's the answer to X1 x2? -2. What's the answer to F x2? Actually -1, and actually smaller. I'm done with these values. This function is definitely an increasing function. Now, immediately after that, I think I should definitely give you this function, starting from my linear reference function. I'll give you, Professor, again, the function F from real numbers. Define the real numbers and draw the graph of the function FX = -x. Now we have an idea of ​​how to draw it. We say, let's first make a value table. If you make a value table, you can't draw the graph of a linear function. No, friend, it shows negative infinity and positive infinity. If I give x 0, the function's value is 0. If I give x 1, the function's value is -1. Look, if I give 2, the function's value is -2. If I give x -1, actually put -1 instead of x. Look, the answer to F -1 is -. From -1, the negatives cancel each other out, +1 comes. Put -2 instead of F, the answer is +2. Professor... So, can't I create the sign table of this function by looking at the value table I made above? Yes, I can. Remember what we were looking for? We were looking for the line where the function's value is 0. By the function's value, I mean F from x, okay? y'LD y = FX. You look at 0a 0da. Now, write x = 0 here and put a bar below it. Look, the function's value is 0 here. So, what values ​​did the function take on this side? To the right of 0, it's minus. To the left of 0, it's plus. Ah, this function takes plus values ​​on this side and negative values ​​on this side. This means that the FX values ​​are less than 0 in this range, and the FX values ​​are greater than 0. Now we come to the most critical point, the second angle bisector line. This is the line we will draw. It's useful to know it by heart. We draw it using the reference function. Actually, pay attention, I'm going to connect the points. Just which points did we pass through, sir? We passed through the point +1 to -1. Look, there's a point here. We passed through the point +2 to -2. There's a point here. We passed through the point -1 to +1. We passed through here, the other one... I don't need to draw it anymore, I think the function is very clear already. If you know the two points, you can draw it now. The function I drew is y = FX = -x. Okay, let's write the domain of the function. Look, where does the domain start? It starts with the numbers R, and when we say the domain, we look at x. How far does it go? To plus infinity. So the domain is all real numbers. Can you tell me the image set, sir? In the image set, we look at y. Look, it starts at the y line and goes from the bottom to the top. In short, all real numbers. Can you make the sign table of the function? I did. What else was it asking? Let's check from here. Let's enter the function's II function's II. Where is the 0 of the function? Where is the value of the function? That is, when I give x 0, what is the value of the function? It's 0. Great. What else did it want? Sir, we've done the sign of the function. Let's look at the maximum point of the function. Sir, where is the maximum point? There isn't one. Can you see a place at the very top? There isn't one. Is there a minimum point of this function, sir? There isn't one either. Those that exist will come in a bit. Okay? I'll draw that kind of function in a bit, don't worry. That's also valuable for us. Now, is this function one-to-one, sir? How did you understand that? I cut the horizontal line at one point, I cut it at one point, I cut it at one point, I cut it at one point. So what is this function? It's a one-to-one function. It's a one-to-one function. Agreed? We're great, we're great, sir. Is this function increasing or decreasing? Look at the graph, man, the graph doesn't lie. Where is the graph going? Downwards. If it's going downwards, then what is this function? It's a decreasing function. If the graph of a function is decreasing, then... Remember how we wrote the definition earlier? Let F be an element of X1 and X2. If, between the values ​​of X1 and X2, x'L increases while the value of the function decreases, then we call that function a decreasing function. I'm curious about the definition. I'm writing this especially for my friends. I wrote this here because I thought it would be beneficial to see it. Is that all? No, I'm not finished yet. Then, an extra page. Look, we've also drawn what the function will look like when a minus sign is placed in front of it, right? Look, what happened when I put a minus sign in front of my function? It suddenly changed direction. Actually, do you know what I did? When I put a minus sign in front, all the y-values changed their signs. So, in short, what did I do with the function? I took symmetry with respect to the y-axis, right? Was my function like this before? y = x. Taking symmetry with respect to the y-axis means moving this part here and that part there. For now, let's wait, it's coming, I'll explain it to you slowly. Don't worry about the three, I'll take symmetry with respect to the x-axis. I said it wrong, I drew y = x. y = -x means taking symmetry with respect to the x-axis. Look, I took symmetry with respect to the x-axis, this went to this side, this part went up. Anyway, don't worry about that part, I'll tell you about it in the next parts. The first thing I said was wrong, but we corrected it later, sir. What should we know? Look, you should know how to draw these very well. The question is, my function is again about real numbers, real... You should know how to plot FX = 2x, GX = 1/3, and hx = -3x, all defined on numbers. Why am I emphasizing these specifically? I want to write a special note about them. Now, my functions are linear reference functions, what was it? This is where it's derived from y = x. The value M here is the slope of our function. Understood? If my function could be an increasing function, or a decreasing function, if it's an increasing function, its slope will be greater than 0. If my function is a decreasing function, its slope will be less than 0. And it's always like this in linear functions: If the value M is greater than 0, look, I'll draw it like this, so that this part is an acute angle. Pay attention, if there's an acute angle, A, they're all increasing. Also, if I draw the function as an obtuse angle with the x-axis, look, this is really an obtuse angle. If I draw it as an obtuse angle, what happened? Pay attention, it became a decreasing function. And how do we know this? By looking at the coefficient in front of it. There shouldn't be a problem up to this point, very quickly. Now, how can I draw this graph? I can draw it using the linear reference function. First, I draw the linear reference function. What was the linear reference function? y = FX. That's equivalent to x. Actually, even though we don't see it, what coefficient is in front of X here? There's 1. If the coefficient in front increases, my function moves to a steeper position, it becomes like this. For example, y = FX = 2x. Why did the slope increase? Now the function will grow faster. To grow faster, it moves to a steeper position. Understood? If my function's graph has 1/3, we drew y = FX in front of it, right? Yes, sir, we do n't need to have a graph. y = F, what did we call this? We called it the g function. GX = 1/3 x. Great! Remember I mentioned another function earlier? I told you there's another one, the second one, also known as the angle bisector function. Let's say y = MX. That's equivalent to -x, right? If the coefficient in front of it is even smaller, both negative and smaller, where does it have to pass through? Look, again... It's shifting more quickly, taking on negative values, smaller values, and consequently, it's moving to a steeper position. Its value is written as H = x = -3x. Did we agree, Professor? I want another function. Would I ever disappoint you? The KX function is equal to the -1 2x function. So what will that function do? It will be a bit more horizontal, but on the negative side. Knowing how to draw these is very useful, but I'm drawing a rough sketch. Even if I draw it, it will definitely look like this, there's no way around it. Honestly, we've created quite a colorful position here, but at least I quickly explained how to find any linear function without any plus or minus signs, using a linear reference function. Look, this function is increasing, this function is increasing, this function is increasing. But this function is decreasing, this function is decreasing, this function is decreasing. None of them have maximum or minimum points. No, Professor, isn't there anything that does? Of course there is. Let me give you an example of that right away. Look, I'm repeating everything again. F = this time -3, from -2 to 3, towards the numbers. Let's draw the graph of the FX = x function together, Professor. What's wrong? We'll draw it here, right? We just learned what we were looking at. We were looking at the value table, weren't we? It was very easy to draw using the value table. Because I wrote x again, I wrote F x again. Pay attention, this time there's no infinity, no plus infinity. The range between -2 and -3 is closed. The range is -2, -1, 0, 1, 2, 3. If the range is very large, for example, if there are many numbers here, use dots in between, but it's also useful to write the last number. Agreed? You absolutely must see the last number. When I give -2, the answer is -2, when I give -1, it's -1, when I give 0, it's 0, when I give 1, it's 1, when I give 2, it's 2, when I give 3, it's 3. Now I'm going to draw the graph. This is where it gets interesting. My friend, it's useful to draw the endpoints of my function first. What are our endpoints, sir? -2 K -2. Let this be our answer. Again, what do we get? -2. -2 -2 point. This is our other endpoint, sir. The 3x3 point. The point I get when I draw at the 3x3 point. Is this it? Can I now draw the graph of the function completely? CKS determine your endpoint and starting point. The points in between will already match, there's no escape. For example, where did the answer -1 go? -1 came. Where did the answer +1 go? +1 came. 0 to 0, right in the middle. What did the answer 2 to 2 come to? It came here. My function came here. Under normal circumstances, if I were to draw the linear reference function, I would say, "Look, the graph of this function should continue from here. Let's draw it in a different color so it doesn't get mixed up. The linear reference function should continue infinitely from here, but it doesn't." So, the domain of my function has changed. Where did the domain start, sir? Who were we looking at in the graph? The X line. We started from -2, where did we end? The graph ended at the 3 line, sir. Ah, from -2 to 3. Okay, let's write the image set together. Image set. What happened? We're looking at the y lines. At the very bottom, -2, at the very top, +3. And the image set of this function is also the same way. Did it become -2 to 3? Yes, it did. Let's continue. What else were we asking? The image set is the zero of the function, sir. There was no change in the zero of the function. The zero of this function is definitely where FX = 0, where X is 0. You said "place," I put a period here, there was no change in the function's table either. Look, I wrote x, I wrote FX. Where is the place where X is equal to 0? The function is positive on this side. Look, the FX values ​​are really positive on this side. How do I understand? The graph is above the x-axis, this is very valuable. Where is the place where the graph is below the x-axis? Look, is FX less than 0 here? The decreasing limit didn't affect this function. What is this function still doing? It's going up rapidly, so it's an increasing function. Okay, we haven't found another maximum point of this function, sir. Where is the maximum point of the function? The maximum point of the function means the place where the function takes its greatest value. What is the coordinate of this place? 3 by 3. So, I've arrived at the maximum point of this function, 3 by 3. If it had said "maximum value," look, the word "value" always reminds me of y. What is the value? The maximum value, you would just write 3. Okay, sir, when you say "where is the minimum point of my function," it's the lowest point, the lowest coordinate, pay attention, what is this coordinate? -2, -2. So, -2, -2. If it had said "minimum value of this function," I would have looked at the y value. What is that? - 2, my dear friend. These are the types of functions we need to know, at least initially. I assume we've learned them like a pro. Now, let's continue. We'll look at the types of functions we derive from linear reference functions, such as ax, FX - + r + k. You might say, " What does this have to do with our function?" Let me show you. Our question is: Do you know where we get that function from? Again, let's define the function from real numbers to real numbers. You know the FX function equals x. My friend, where I see x, I got bored and said, "I'll write x + 3." Agreed? Okay, when we write x + 3, we get a new function. Let's call it the GX function. What should we write where we see x? We said, "Let's write x + 3." Actually, what I did was the FX = x part. How much did we add to the FX function? We added 3. Great. First, let's draw its graph, then I'll explain the logic and talk about translations. Don't panic. First, how do we draw the graph of a function of the form x + 3? Let's look at the sign table. Of course you will do it. I apologize, I showed the x function on this value table, I showed our limits, I showed the extra length. First, what am I trying to find? The value that takes the answer to 0 makes my job incredibly easy. Look, for x + 3 to be 0, what value should I give to x? -3. So I say, if I give x -3, what is the value of the function? 0. Let me increase it a bit. If I give x -2, in the g function, if I give x -2, what is the answer from -2 + 3? a becomes 1. If I give x -1, in the g function, what is the answer from -1 + 3? 2. a becomes 2. Okay, let's give -4 and -5. If I give -4 in the g function, what is the answer from -4 + 3? -1. Again, if I give -5 instead of x in the g function, what is the answer from -5 + 3? -2. Is there a problem up to here? No. One of the most valuable parts for us is where the function's value is 0. You don't forget this part. I said I can draw the graph. Come on. Let's draw the graph of our function together. Don't panic! I'm sure you 'll do great on this exam, or functions. Actually, we're solving quite a lot now. Look, where do I start? -3 0 point -3 This is -3 0 This is -2 1 point. If I take -2 here and -1 here, we'll pass through the point, right? -1 2 point. If I take -1 here, what will this be? It will correspond to 2. I showed it like this. This is the 2 point. I can draw the other points too, but anyway, let me draw them. One more, where do we pass to -4, sir? We'll pass from -1. Look, at the place we call -1, we passed from here. Then I can connect these points, even drawing just two would be enough. What is this function I obtained? You can start with y equals every time. GX That equals what? It's the graph of x + 3. Now, let me quickly explain it to you. What is the coefficient in front of X? 1 positive. So, can we say that this function is definitely an increasing function? Yes, we can say it's an increasing function. Okay. Come on, if I were to make the sign table of the function right away, where is the value of the FX function on x? It's 0 at -3, so the value of the function is 0 at x = -3. Look, right here, right at this point, wait a minute. Let 's take another pen, right at this point, it's 0. Where am I on the x-axis now? Where is it? To the right of -3, so the function takes values ​​greater than 0 here. Really, for God's sake, can you look at the graph? Look at the graph, it's on the x-axis, and where is it, sir? It's under the x-axis, did you see? If it's under the x-axis, in this region, what will we say there? FX is less than 0, sir. Look, here it's greater than 0 plus, here it's less than 0, the value of this function is definitely finished, that's it. Increasing or decreasing. You can decide even before drawing the graph by looking at the coefficient in front of X. What more are you saying, sir? If I print the domain, it's very easy, we've been writing it for a while now. Look, the range, the domain, the widest range, sir, the real numbers range. Can you print the zero of the real numbers function? Ah, look, it's different from the reference function here. Where did we cut off? - In section 3, look, since the answer to F-3 is 0, x = -3, my function is 0, the zero of the function. Also, pay attention to where I use it; we use it to represent 0 in the sign table. This is also important. I'm almost done, I'm really tired, but I'll continue. When you ask about the maximum point, I say, "There's no maximum point, no peak value." When you ask about the minimum point, I say, " There's no lowest point, it just keeps going down." Is this function a one-to-one function? Yes. When I apply the horizontal line test, I only intersect at a single point each time, so you'll say this function is one-to-one. We've stated the range where the function is increasing or decreasing. More precisely, what is this function always? It's an increasing function. By the way, there are also functions that aren't always increasing, I should tell you that too. Now we'll draw another function, and then I'll try to connect the topic to something else, translations. I don't want to finish these lessons without mentioning translations. There are also absolute value functions. I'll explain those too. We'll finish all of them in this lesson. Let me see how many minutes it's been. Wow, keep explaining, teacher, keep explaining! I really love explaining! Hopefully you can understand too, how happy I am! So, let's take another function F, from real numbers to real numbers, this time let the rule of the function F be FX = -2x + 6. I want to draw its graph. What did I say? I said, if you first find the value that makes it zero, your job will be much easier. Do this immediately, you'll say it in the exam, Professor Mehmet said, okay? -2x = -6, what should I give to x? 3. Then what do we do? We said, the sign, sorry, the value table. If you do the value table, drawing the graph will be easier. I said x, I said FX, this is negative infinity, this is positive infinity. What is the value that makes it zero? -3. Look, where do I write -3? I wrote it here, a little to the right. Just before it was -3. Should I give something different? It's writing questions off the top of my head now, let's not make it like this. Let's give 4, let's give 4. Let's give 4 here. I said +4, it came out as -4, x = 2 came out. I'm trying to make it different, but it was +2, right? It was -3, it was +3 actually. Anyway, like this... Let's say +2 came, sir. When I gave +2, what became the value of the function? It became 0. You can try it if you want. Look, when I wrote 2 instead of x, -2 x 2 + 4, the answer is 0. Now I increased it by one, I gave 3, -2 x 3 + -6 + 4, what happened? The answer is -2. Let's give one to the left, it doesn't matter, you gave too little. Sir, too little, it's not too little, it's enough for us. You won't say too little, you'll manage. Okay, friend, what did I give instead of x? I gave 1, what did I get? 2. Now let's see, even two points would have been enough for the graph of the function, but I have extra points. I said, "More is better than less," so I sat down and started drawing. I determined the positions of the graphs, but I'll shift these a little to the left because we're going to do operations now. 2 passes through the 0 point, let's say point 1 is here, point 2 is here. This is exactly. So it will pass through this circle. 3 passes through the -2 point, let's say 3 is -1 and -2. Friend, it will pass through the -2 point, right? Okay. 1 will pass through the 2 point, point 1 is here, did it go directly to 2? Let me see, -1 went to 2. Okay, there was no mistake. Where does it pass through? It passes through point 2 with the function. Okay, now I can draw the graph. What is the graph of this function I drew? The graph of the function y = FX = -2x + 4. And tell me immediately, is this function increasing or decreasing? Of course, decreasing. If you say anything else, you'd be really wrong. Look at the graph, it's going down loudly. I do n't need to look there, sir. If this is negative, it's decreasing. That's it, that's it, that's it. Come on, let's make a sign table. M, where were we looking when we were making the sign table? Actually, it's where the function intersects the x-axis, or look, it's the point 0a 2, right? At x = 2, the function's value is 0. But to the right of this, different things can always come. Look, for example, to the right of this, I'm below the x-axis, so my function took negative values. Really, look, it takes negative values ​​to the right and positive values ​​to the left. Really, sir? Show us so we know. Really, look, is n't the graph of the function above the x-axis, on the left side, to the left of x = 2? If this side is above the x-axis, the function's values ​​are positive. I've shown those positive values ​​here, and the sign table is also important for us. What else do we need, sir? In a function, the domain is real numbers, the range is real numbers. I haven't set any limits, so there's no problem. We'll especially need these domain and range sets in absolute value functions, which I'll explain shortly. We've finished this, sir. What else can you give me? What can I give you, brother? We talked about the domain, the maximum point, there's no maximum point because it goes to the top. There's no minimum point either. What else, sir? Where is the zero of the function? F2 = 0, so x = 2 is the zero of our function. We said the function is decreasing, we made the sign table of the function. Again, like the others, what can we say about this function? Is it a one-to-one function? Where can we say that, sir? Look, I'm doing a horizontal line test. How many points do I intersect at each time? 1. When you intersect at two points, or three or more, are you cutting it off directly? You're saying it's not a one-to-one function. Agreed? Now, this too... Now that we've learned how to plot it, let me start by giving you a general explanation. If you're given the FX function – any function, it doesn't matter – and you add a number outside the function, for example, the number A, this means that on the y-axis, yes, if you add a value outside the function, it will move 'a' units up. Of course, this is if A is greater than 0, okay? Okay, sir, if I write FX - a in the FX function, it means I'll move the graph down 'a' units on the y-axis. You'll move the graph down 'a' units compared to the initial graph. Okay, but what if I add or subtract something from the function? What if I do FX - 1? Let's not say -1, what if I do x - a? By the way, I didn't assume 'a' is less than 0 here; I assumed 'a' is greater than 0 because I'm subtracting, okay? Let me give you a number here so you don't get confused. For example, I wrote x - 3. This time we'll be working on the x-axis. If you're doing something inside, you're adding something outside to the x-axis. You're extracting it on the y-axis. Here, you're doing the opposite. When you say -3, you move it 3 units to the right. Normally, when you move to the right, the values ​​increase. Here, you're acting as if -3 is increasing. Similarly, if I ask you about the graph of FX TX + 2 on the x-axis, would you shift it 2 units to the left compared to the initial situation? That's the whole point. For example, I have the function y = FX, and its equivalent is the x-axis function. I called this function FX, right? If I ask you to draw FX + 3, you'll say, "Can you draw it very quickly?" You'll say, " Sir, where should I shift it 3 units? Upwards." This function was here, where it intersected the x-axis and the y-axis. I shifted it 3 units upwards. What happened to the function's position? Look, I have to draw it parallel to the other one because it's linear. Now it will pass from 3 to -3. We don't have to spend time drawing the function; we can draw it instantly thanks to these translations. Of course, I'm not focusing too much on the translations here. I 've given all the necessary information about these. Now we can move on to the graph of the absolute value function. The graph of the absolute value function – this is really important. Guarantee This is one of the places where we expect questions, because it's much better to ask about the domain and image sets here. I want to know if you can draw the graph of FX = mlak X for me. We'll find the domain together. How will we draw it? First, you act as if there is no absolute value. Look, Step 1: Let's proceed step by step together. Draw it as if there is no absolute value. Draw it as if there's no 'r', draw it as if there's no 'r'. Okay, what does this mean, sir? If there's no absolute value, we call this function y = FX = x, right? Do we know how to draw it? Let's say we don't know, what would you do? You'd say, " Sir, I'll immediately make a value table for it. When you give 0 for x, the answer is 0. When you give an answer for x, the answer is 1. When you give 2 for x, the answer is 2. When you give -1 for x, the answer is -1. You can increase the number of points." Then we draw the graph of the function. Look, it's an incredibly good tactic. We got x and y. We got it. What is the graph of the function I'm going to draw? This is it: the graph of y = x. It's a bit messy, y = FX = x. But when you say you want the absolute value, you take a symmetry upwards from the part below the x-axis. You have to do it the same way, but look, I took a symmetry upwards. Sir, what do we do with the top part? If it doesn't touch the top part at all, then erase the bottom part as well so it doesn't get mixed up. What graph have I drawn? I drew the graph of y = FX = FX. When it says absolute, we'll move the part below the x-axis upwards. If the part is on this side, I'll move it this way; if the part is on that side, I'll move it this way. That's the logic. Now I've come to the important points. First, what will we look at? We'll look at the widest domain. Let's write it together. I've got my white pen again. The domain equals, brother, the domain is the region I've shaded on the x-axis. Look, where do I start with x? From negative infinity. I've moved along the x-axis with the graph, and where did I go? To positive infinity. How did we represent that? We always represented it with real numbers. I came to the range. Where did the range start on the y-axis? Look, 0. This is it, brother, 0. I started from 0. Where did I go? Look at the region above the graph on the y-axis. Look, look, look. Where did I go? I went to positive infinity. Therefore, its domain is the closed interval from 0 to the open interval of positive infinity. Look, it's different from the other questions. That's why it's important. So, where is the 0 of this function? 0 means where it intersects the x-axis. Since f0 = 0, x equals 0. My function's 0 is 0. Can you make the sign table of the function? Do it, we'll even tear it apart. Come on, I wrote x, FX. Look, I also wrote mlak x here. We'll write, when I give x 0, is this where the function's response is 0? Yes, the function is positive on the right side of 0, and positive on the left side, also positive, sir. So, pay attention, does the function always stay on the x-axis? It always stays on it. Super, look at what we're solving! Okay, we'll examine the interval where the function is increasing or decreasing. Pay attention, what is the function in this interval? A is decreasing, sir. Okay, in this interval, where does the function's graph go? It goes increasing, sir. Wow, incredible information! Why? I'm coming from positive infinity, up to 0 on the x-axis, what did I do? I behaved decreasingly. Okay, I started from the 0-axis, up to where? Up to the positive infinity-axis, I behaved increasingly. Please don't forget to keep these closed, don't just memorize. Let 's write down the maximum and minimum points now. We're solving this so well, we're solving this so well! I'm incredibly enjoying this lesson right now. I hope it's the same for you. Maximum point... Brother, can you see the maximum point of the function? I honestly can't see it, it doesn't exist. But if someone who does it by memorization says it doesn't exist, they'll fail. Minimum point is clear, are you going down further on the y-axis? No, you're not going down. Which point? That's 0a 0. The point is 0a, the point is 0. If it asked like this, if it said the minimum value is at 0, so at 0a, the point 0, the minimum point, you write the y value you wrote there, what is that y value? 0, sir, the minimum value is 0. Is it 1E to 1? 1E to 1? Here's another critical question. What did we say? Horizontal line test, look. I did a horizontal line test, we cut at two points, sir, at two points, sir, cut right from here. I cut at a point below, it doesn't matter, even once. If you cut at two points, this function is not one-to-one, it's not one-to-1. Okay, if I asked you to show me an example, for example, what is the answer of f1 because of the absolute value of 1? It's 1. What is the answer of f-1 because of the absolute value of -1? It's +1. In normal conditions, are x1 and x equal to each other? No, but what did their values ​​turn out to be? They turned out to be equal, sir, both of their values ​​came out to be 1. That's why this function is not a one-to-one function. You have to find a different x value for each different x. That's the biggest joke, brother. Let me write you another question about absolute value, even a little bit. We're going to try to solve a simple question, okay? Try to make sure there's no problem here, otherwise I've solved a zillion examples related to this. Please, please, please, you need to come and watch those written practice videos for the "Question Hunter" series. We solved a lot there. Of course, you also need to watch the functions lessons, sir. Our teacher asked about the graph of x - 2 + 3 in the exam. There's a very fast method for drawing it, we'll start with the long method first. Oh, and let's not forget to mention this above: FX = the function x inside the absolute value. Every absolute function is written as a piecewise function. What is the value that makes the inside 0? x = 0. If x were greater than 0, that is, if the sign of the absolute value were 0 or greater than 0, it would come out as is, or if the absolute sign were 0 or the x value were less than 0, it would come out with a changed sign. Therefore, we can take this absolute function out in two ways: either x comes out as is when x0 is equal to B, or it comes out with an addition before it when x is less than 0. Now, this is our example, I 'll come back to that example, but this was important, I'm skipping that part too. I don't want to, sir. Our teacher asked something like this on the exam: He gave the function 2x - or let's say +8. He asked if we could write this function as a piecewise function. Either first set the inside of the function (the absolute value) to 0, or the whole function, it doesn't matter. 2x = -8 came out. The answer for X is -4. So, if this function takes values ​​greater than -4, it comes out exactly as it is. "Exactly as it is" means 2x + 8 is the definition of absolute value. I explained this in absolute value. For those who didn't understand this part: if x is less than -4, change the sign and come out. What's inside? 2x + 8, change the sign - 2x - 8. So, the rule for the FX function is that it comes out as either 2x + 8 when x > -4 or - 2x - 8 when x < - (sorry, I can't put a negative sign anymore, it comes out when x is less than -4). Now, let me get back to my question. I have this function. What do we do? First, we'll try to draw the answer of this function as if there were no absolute value. That's not right. Normally, you'll imagine this function like this, of course. Look, we'll think of it as if only this much exists, then we'll try to do the rest. Agreed? It's solved using the translation method. If there's no translation, you'll say, "Sir, what's the value that makes the inside of this part zero in my function?" We'll draw this as a piecewise function. The first way, look, let's draw it long from the first way. First, we'll convert it to a piecewise function. Let's write it down: x - 2. What's the value that makes it zero? x = 2. If my x value is greater than or equal to 2, how would this expression come out? It comes out exactly like that, x - 2, and next to it is + 3. The rule of the function is: What happened? x + 1. If x is less than 2, this function changes its sign and comes out - x + 2. What's next to it? + 3. The function becomes - 2x. Sorry, - x + 5. In short, the new rule of the function is that when x > 2, the function becomes x + 1, and when x is less than 2, it becomes - x + 5. I'll continue from below, I'll draw the second way next to it. Now I'm starting to draw the function, sir, but I'll make the sign table. Or, I'm like this... Some people might say they can't see it, so let's solve it right away in the sign table. I'll create two sign tables for x and the value of x + 1. Look, first, what is the value that makes the inside of this function zero? It's -1, right? Look, I gave -1, the inside became 0. I gave 0, let's go this way, it became 1. I gave 1, it became 2. I gave -2, the answer is -1. Great, we'll connect these points and write it down in a moment. Now let's look at the second function, -x + 5. What is the value that makes the inside of this function zero? If you give 5 instead of x, the value of the function becomes 0. If you give 6, the value becomes -. Okay, let's give a smaller value, what will the answer of the function be if I give 4? It will be +1. Now let's show these points here, but remember where the critical point is for us: x2. First, draw a line at the level of x2. Okay, to remember this, the level of x2 is here. Then you draw the graph: 0. Wait a minute, where am I? -1. Where is 0day? Here. -1. At 0day, I'm at 1. At 0day, I'm at 1. I'm here, sir. The alignment of 1 to 1. This is where we are at 1 to 2, sir. The place we call 1 to 2 is here, it was here, let's extend this a little upwards, the line of x = 2. I will draw this graph when x is greater than 2. Where is x greater than 2? Look, this side is where x is greater than 2. So I don't need this part, I need to delete this side. What am I saying, friend? This is the graph of my function, okay? I don't want this side, I deleted that side, so I said I'm only interested in that. It's not over, our work continues. Okay, I said I would draw a difficult graph. Here is its graph. We will pass through the 5 to 0 point, sir. Where should 5 to 0 be? Let's say 5 to 0 is here, I marked it. Sir, we will pass through 6 - 1. Let 6 be here, -1 is here, and 6 - 1 is here. Sir, where will we pass through 4? We will pass through 4 to 1, sir. We will pass through 4 from here, sir. Okay, I'm curious, where will the graph below pass from the alignment of x = 2? It's also worth noting that if I write 2 instead of x, where will the graph of the function -2 + 5 pass through? It passes through 3. If I write 2 instead of x in the function above, it becomes 2 + 1. Where does it pass through? That also passes through 3. Actually, the graphs of both functions pass through this line, right? If my function were to pass through here, it could pass through here. Now I'm drawing the graph of the function. Let's say it's almost there. Let's say this point is here. I'll draw it like this, but we don't want this point. We want the points where x is less than 2. The graph I need to draw is only the green graph. Do these graphs have to be drawn long and detailed every time? Absolutely not. I'll start with the graph of the reference function. Look at the situation: we 've drawn the graph of y = X, right? y = FX = x. What's next to it? -2. Putting -2 next to X means shifting it 2 units to the right. Now I've shifted that function 2 units to the right. It looks great: y FX = x - 2. But should we draw these separately? I'll draw them step by step, 1 step by 2 steps. You should definitely do this in the exam too. Now, step 1: The red function above, then what did I do with it? I showed you earlier. Actually, you can see it here. Step 2: Then I shifted it back one step to the side. Step 1: I drew it, sir. Step 1: I drew this too, sir. Now, sir, it asks me to take the absolute value. Look, what's next? Absolute value. Taking the absolute value means what do you do with the part of the function's graph that is below the x-axis? It means raising it. By the way, where is the point where this function zeros out, where it intersects the x-axis? It's 2. Look, this is the part of the function that is below the x-axis. What am I going to do with this? I'm going to raise it. It's that simple. That function was coming before, like this. Now, what happened? It folded up. I raised it. You're great! How many steps is this now? It's 3 steps now. We've reached step 4. What am I doing in step 4, sir? Let's look at the function's graph. What are we doing? I added 3 units outside the function. I'm going to move it 3 units up on the y-axis. Remember, translations too. I explained these to you earlier. I'm drawing the x-y coordinate plane again. I drew it. This is the x-y coordinate plane. Look at this coordinate plane. From the line of 2, the graph is now... What happened? Ah, sir, this is the graph of my function. With absolute value, the function I obtained is now the graph of y = FX = x - 2 + 3. That's it, it's perfect! The graphs of absolute value functions can be drawn like this, my dear friend. And finally, of course, I won't go into this anymore, I'll just explain this from here using the control point. But if you want, sir, to draw graphs related to inequalities or anything else, I'll do my best in a different video. Because we've passed an hour. I think this is enough. Now, MX + n = 0, the functions we've been drawing since earlier... Actually, how did we do it? We said, sir, let's say there's a function like MX + n. First, we zero out this function. When we zero it out, look what the x value becomes: -n/ m. What does -n/m become of this function? It becomes 0. -n/m is the 0 of this function. On this side, we decide whether the function is positive or negative. Remember, in these functions, if I have two real functions like F and G... If the numbers have linear functions defined as such, the solution ranges for inequalities FX < GX, FX < GX, FX > GX, FX > GX will be as in this table, but these tables are for memorization. Let me briefly explain the logic to you, sir. If my FX function is drawn like this, and my GX function is always below it, the most important feature you can write about these functions is that if we look at their graphs and make a comment, the function whose graph is higher has a greater value. Look, I should be able to make this comment here, or you should be able to make this comment by looking at these graphs, sir. In my graph, this time the GX graph is higher, for example, like this, okay? We did it like this, what was this, let's write it with the GX graph, what is FX trying to tell us here? It says whose graph is higher, the GX graph is higher, whose graph is lower, the FX graph is lower, that's it, which one has a greater value is clear. The most critical question types are: if I draw a GX function like this, if I draw an FX function like this, look, look, look, look at the situation, let's say the intersection point of the two lines is at x = a. By the way, the intersection point of the lines is exactly... It is shown by equality, meaning the points where FX and GX are equal are found by equating their equations. So, right after this equality, I see who is on top, I see that GX is on top. In this region, GX is larger than FX, why is the FX graph lower? Okay, let's look at what's happening in this region. GX's graph is below. GX's graph fninternbasvuru is amazing, what more could you want? What more could you want? Just being able to solve these is enough. Okay, sometimes in questions we'll ask, "Can you describe this region to me?" Who is at the top? GX is there. Where is that? Isn't it the green region? The values ​​in the green region are greater than the red ones, meaning FX is greater than GX's values, but smaller than GX's values. That's the whole description. We solve different types of questions with this. Okay, "Can you describe the yellow region to me?" Let's describe the yellow region. I went down, whose graph am I below? I'm below the FX graph. So, if you're below the FX graph, I'm smaller than its values. The values ​​in the yellow region... "Okay, sir, in inequality questions, I've always solved my problems like this. Go look, I'm bigger than GX's values, but that's the essence." Solving problems is a separate matter. This is the theoretical part of the job, the mathematical aspect that needs to be known. I'm sure the top-ranking students are having incredible flashes of insight into what I've explained. I even have something very special to tell you: If I consider this line equal, in this region, GX and FX are greater than or equal to each other for a moment. If I consider the line equal, look, if I don't take the line, it's greater than or less than. If I consider the line equal, I can write down where it's equal. Look, in this region, the FX graph and GX are both greater than each other and equal once. This is the truly theoretical part of the inequality. I tried my best to explain all this to you for 1 hour and 5 minutes. I hope it was understandable. At least before the exam, even if it's multiplied by 2, I'm checking to see if I've really shown you all the points I wanted you to know about function graphs. Come on, come on, one more thing, last one, last one, I swear, Mehmet teacher, he can't bear to leave it out. I'm afraid to leave it out. Let's look at the domain of this function that we just drew. Look, these are important. Where is the domain? Look... It comes from negative infinity and goes to positive infinity on the x-axis. So, all real numbers are critical in the image set. Look, where is the image set, sir? It starts at the level of 3. When you say image set, you have to look at where it goes. It goes to positive infinity. So, its image set starts from +3 and goes to positive infinity. What else were we looking at? The image set, the function's 0. Where is the 0 of this function? This function doesn't have a 0. Because the 0 of a function means the point where the function's value is 0. Pay attention, the smallest value of this function is 3. How could I forget such an important thing? Oh my God, this function doesn't have a 0. A function cannot have a 0 unless it intersects the x-axis, okay? It has to intersect the x-axis. I think I made the sign table, I didn't. Let's do that too. Let's make the sign table. The function's 0. No, sir, where will the sign table be? No, this function always takes different values. Look, whatever you write in place of x, what are the signs of the function? They are always positive. So, that's it. Man, this is a very positive function. Okay? Write the interval where the function is increasing, decreasing. Write the range carefully, pay attention to the x = 2 line. Look what's happening on this side, it's behaving in decreasing order. Isn't my function a decreasing function? Which range? From negative infinity to +2? The range from negative infinity to +2 is decreasing. The intervals are definitely closed, the decreasing order is done in a closed way. Look what it's doing here, it's increasing. Where does it start? It starts from the 2 line and increases until the positive infinity line. My function is definitely here. I said it's an increasing function, I said it's increasing. I came to the maximum point of the function. Can you see the endpoint above? No, then the function doesn't have an endpoint. Where is the minimum point of the function, sir? It's clearly visible. Look, the minimum point is the bottom of the function, the bottom point is the x value 2, sorry, not a closed range. We write it as the x y value in the coordinate plane, the 2 by 3 point. Look, this is an open range. Ah, the 2 by 3 point. Okay, I wrote it as a point there. I'm looking to see if there's anything else left. I'm really tired, my head hurts a lot, but I'll finish it, God willing. We can look from above. Maximum, minimum point, one to one, one to one left, sir. Tell me one to one, is it 1 to 1? This isn't a function. How do you understand that, teacher? I did the horizontal line test. Look, I intersected at two points. If it intersects at two points even once, that's enough. This function isn't one-to-one. Oh, I almost forgot, please add this to your notes before you leave. Sorry to those who left early. You shouldn't leave early. You shouldn't leave early. It's not over until Mehmet teacher says it's over. Now it's over. But that's enough for now. Let's talk later. 1 hour and 10 minutes is enough, my dear friend. If you support me with your comments, if you write what you want in the comments, Mehmet teacher will continue to explain as long as his throat allows. Take care, stay well, goodbye.