 In our previous video, we mentioned that if a function's one-to-one, it has an inverse function, and as the inverse function switches the roles of domain and range, if you want to find the range of a one-to-one function, just compute its inverse function and find its domain because the two swap places there. Unfortunately, most of the functions we study in a course like pre-calculus or at our college algebra, these functions are not one-to-one. So the previous method of using inverses to find the range of the function has limited use, right? We have to be a one-to-one function. Honestly, the best way to find the range of a function is really just to explore its graph. If we know the graph of it, then we essentially know the range because we can see it on the graph. Now, there are things we can do without necessarily pulling on a graphing calculator to analyze these things from a purely algebraic point of view, but essentially the graph is where we want to go. For this reason, I should say that we've drawn so many different graphs throughout this lecture series, knowing the graph tells us so much about the function. The simplest graphs are those which are transformed from standard graphs. So some of the standard graphs that we've seen this semester, we have various power functions, right? So power functions, this includes things like y equals x to the a, where a could be some type of integer positive integers. We get monomials. It could be a negative integers. We get the reciprocal functions. It could be a fraction. So we get like radicals and then combinations of power functions and radicals together. So power fun. And then I mean, a also could be some irrational number. So there's a huge variety that we can see when it comes to power functions. The other big family that we have to pay attention to is the exponential family, exponential functions, in which case that's a function that looks like y equals a to the x, where your base is allowed to vary. It could be a growth model. It could be a decay model. We can also take inverses of these functions that gives us logarithms. So y equals log base a of x. We should know that graph. Now the inverse of a power function is likewise a power function. So we don't need to separate those two cases. And you know, we also have like the absolute value. That was one we paid a lot of attention to the absolute value y equals absolute value of x. These are some of the basic graphs that we have learned this semester in this, in this lecture series, I should say. And if we can transform these things, then we actually have a good variety of graphs that we could consider. Also, if you equip yourself with trigonometric functions like sine, cosine, tangent, you can basically all of the basic families of graphs that one should know in pre-calculus. So power functions, exponentials, and logs are our main focus. So how can we transform those functions? All right, so given a function, the function's graph y equals f of x, the different types of transformations we can do are things like the following. We can multiply the whole function by a. We could divide inside of the function by b. We could subtract from x and h or we could add to the whole function k. And so this gives us two categories of transformations, the the vertical transformations and the horizontal transformations. And it's important to distinguish between these ones. The vertical transformations are going to be those things that occur outside of the function f, right? You have this format, the vertical things that we do outside of f. The stuff that happens inside the function f, this is the horizontal zone. And things in the horizontal zone kind of work opposite to what you expect, which is why we always do the twilight music when we talk about the horizontal zone. Because it's not what you expect. It kind of works backwards. And so between the vertical and the horizontal transformations, there are three families of transformations we pay attention to. So the first family is reflections. So we could be reflecting across the x-axis or the y-axis. If we're doing a vertical reflection, that means that we're going to reflect across the x-axis. So you have your x-axis here and you reflect across it for that. We call that a vertical reflection because it's the y-coordinate that changes when you reflect downward, right? Y will become negative y and negative y will become positive y. That's the reflection that happens. And that's why we call it vertical reflection. The horizontal reflection is actually the reflection across the y-axis. So we think of it this way. Here's the y-axis. Here's our point x comma y. When we reflect across the y-axis, we're going to get the point negative x comma y. So it's the x-coordinate changes. Vertical transformations change y-coordinates. Horizontal transformations change x-coordinates. Okay? Now the way that we recognize these reflections is that you're going to have some type of multiplication by a negative. If that negative sign is outside of the function, there'll be a vertical transformation reflection across the x-axis. If that negative sign is inside of the function, it'll then cause a horizontal reflection that is reflection across the y-axis. Now of course, if you reflect across the x-axis and the y-axis, this has the effect of reflecting through the origin, like 180 spin. Another type of thing, another type of transformation that we should pay attention to is the idea of scaling, so we could stretch a graph or compress it. Stretching suggests that it gets longer, longer, longer. Compressing means it gets squished smaller, smaller, smaller, like shredder's helmet in the trash compactor from the Ninja Turtle movie, right? So how do you detect a scaling type transformation? Well, this is going to be multiplication. Multiplication outside of the function represents a vertical scale. Now you could do multiplication or division. That is, if you multiply by a number greater than one, that'll vertically stretch the graph. If you multiply by a number less than one, or just divide by that number, right, then that will actually cause a vertical compression. Now we also see that to do a horizontal stretch or compression, we're going to do multiplication inside of the function. Now if you divide by B, that actually causes a horizontal stretch. And then if you multiply by B, that actually does a horizontal compression. So it works backwards, right? So multiplying makes things get bigger in the vertical zone, but in the horizontal zone, multiplying actually makes things get smaller. It seems counterintuitive, but that's the nature of how these transformations work. Okay, now one thing I should mention is that when it comes to functions, we kind of ignore this one a lot, because when it comes to, say, the power functions, you can still see them on the screen right there. When it comes to power functions, we notice that power functions have the so-called Play-Doh effect, that when you have, like, this ball of clay or Play-Doh or Gack or whatever you like to play with when you were a child, and you start to squish it horizontally, like you do some type of horizontal compression, well that ball of clay is going to start oozing upwards, right? So when you start horizontally compressing it, that's equivalent to a vertical stretch. And likewise, a horizontal stretch is equivalent to a vertical compression. So this is the so-called Play-Doh effect. Power functions you think of are made out of Play-Doh. A horizontal compression or a horizontal stretch is equivalent to a vertical stretch or vertical compression, respectively there. So we can often ignore vertical stretching and compressing when it comes to these power functions. Exponentials sort of have that as well, because when you start doing horizontal stretches and compressions to a exponential function, that just changes the base. So it's not, that's truly a Play-Doh. It does have an effect, but you don't necessarily have to worry about horizontal scaling if you allow your base to change. If you don't change your base, then you will have to compensate for those things. The last thing to also consider is the idea of a shifting. Shifting. This is the very shifty one. How do you shift things up and down? Well, if you want to shift things up and down, you just add something external to the function. Adding k goes up. Subtracting k goes down. If you subtract something internal to the function in the horizontal zone, this actually has the opposite effect, right? So subtraction actually moves into the positive direction, which we call right. And adding moves something in the negative horizontal direction, which we call left. And that's how shifts work. Now, to make sure the horizontal zone acts correctly, we have the right order of operations. You do want this to be factored. If you had something like b, let's say bx plus, well, I'll do a specific number. If you had something like 2x plus one, you would actually want to factor this as 2x plus, excuse me, I miss wrote that 2x plus two. If you factor out the two, you get x plus one right here. So this would tell us that we're going to do a horizontal compression by a factor of two, and then you move things to the left by one. So make sure it's factored in order to get the correct transformations. This can be very confusing if you don't factor the horizontal zone. The horizontal zone should always be factored. I also wanted to mention that for logarithms, logarithms as they are the inverses of exponentials, we like, we can consider vertical reflections, vertical, vertical scaling, vertical shifting and horizontal shifting. We don't care about the other stuff for exponentials, but for logarithms, you're going to get the opposites because as you switch from a function to its inverse, vertical transformations become horizontal transformations vice versa. So when it comes to logarithms, we like to consider horizontal reflections, horizontal scaling, and then horizontal shifting, but we also want vertical shifting. You don't need the rest. I mean, you can use them, but it's not necessary. Because it's sort of like a pseudo play-to-effect, not as much, not as prevalent as with the power functions. And so let's then consider how one could graph these type of functions. And then we'll say what this has to do with domain and range. Oh, that's the other thing I want to mention here, that before we get to the examples, one thing I should mention is that when you do a vertical transformation, the vertical transformation only affects the range, because vertical transformations are those that change y-coordinates. The range, the set of all y-coordinates that show up on the graph. So if you do a vertical transformation, you are transforming the y-coordinates of the graph, which could change the range. Vertical transformations always, they can only change the range, they can never affect the domain. On the other hand, a horizontal transformation will change x-coordinates, and the domain is the set of all x-coordinates on the graph. Horizontal transformations can change the domain, but they cannot change the range of the function. So keep that in mind as we go through these examples. So consider the function f of x equals 2 times e to the 1 minus x plus 3. So if I want to graph this via transformations, I need to think of my basic graph. And your basic graph here is going to be y equals e to the x, for which you see its graph illustrated here on the screen right here. This is our basic graph. And so, paying attention to its domain and range, its domain is going to be all real numbers, because there's no number which we can't evaluate e as its power. And the range, though, this is a restricted range for this graph here. The natural exponential, like most exponentials, if there's no transformations applied here, its range is going to be only positive numbers. You'll notice that the graph is asymptotic to the x-axis, it never touches the x-axis, it never crosses the x-axis. Its range is going to be zero to infinity. So what type of transformations are in play right here? We'll see some things. So our function, our function is y equals e to the x. So I want you to think of it in sort of the following way. Our function is y equals e to the box, right? Don't think of it as a variable, think of it as a box. Anything that goes inside this box is going to be inside of the horizontal zone. So the things that are the exponent of e, that's the horizontal zone. The things that aren't exponents of e, turns out those are going to not be the horizontal zone, so those are going to be vertical transformations. So when you examine that, we have this times by two, we have this plus three. So if we enumerate the vertical transformations, what we see here is there's going to be a vertical stretch by a factor of two, and there's going to be a shift up by a factor of three. Those are vertical transformations. Notice how that's going to affect the range of the function, because the other stuff here, the one minus x, this will affect the domain, but this doesn't affect the range. So if we're after the range of this function, I only have to worry about the transformations. You want to go in the right order. You always reflect, then stretch, then move. Those are the order we always go with here. So reflection, there was no reflection. You don't have to worry about that. What does vertical stretching do to the range? Vertical stretching means that we're going to take all the y-coordinates and multiply by two. Well, if you take zero times by two, you get zero still. If you take infinity times two, you still get infinity. So if you just do a vertical stretch, this will not change the range at all. It'll still be zero to infinity. The next one, a vertical shift. A vertical shift means you're going to add one to each and every y-coordinate. The smallest y-coordinate is zero. You add three to that. You get three. The biggest one, which there really is no biggest one, but if you take three plus infinity, that's still going to be infinity. And so we see here that the range of f is then going to be three to infinity because we moved everything up like we see on the graph over here. We can't get below the line y equals three. Now on the other hand, if we think of the horizontal zone, I need to stop doing that. The horizontal zone is going to be affected by this parameter right here. You have this one minus x. This is our horizontal zone. I can't control myself. The horizontal zone needs to be factored though, so we need to think of it as negative x minus one. And so when we do it that way, we can then see what's going on here. The negative sign inside the horizontal zone suggests that there's going to be a reflection. So we're going to reflect across the y-axis. And then the minus one there means it's a shift to the right by one unit. Now what does this do to the domain? Well, reflecting the domain across the y-axis means you multiply all the x-coordinates by negative one. So negative infinity gets times by negative infinity, which then becomes infinity, and then infinity gets times by negative one. Did I say that one right? So negative infinity gets times by negative one, it becomes positive, and then positive infinity gets times by negative one becomes negative. So in that situation, the domain didn't change. The domain will still be negative infinity to infinity, even though it's reflected now. And then if you shift everything to the right by one, I mean, just add one to each point, looking at the end points, negative infinity plus one is still negative infinity, and infinity plus one is still infinity. So the domain didn't change from these transformations. But the final range did. So that's an important thing to notice here. The range is going to be three to infinity. The domain is going to be negative infinity to infinity, like so. And you can see what we did to the graph that summarizing here, we stretched everything by a factor of two. So this graph right here is taller than this one, right? We shifted it up by three, we see that. We then also reflected across the y-axis. So instead of being an increasing growth exponential function, it's actually a decreasing decay exponential graph. And then it got shifted to the right by one. So the original y-intercept moves to the point one comma five. Let's look at another example. Let's consider this quadratic function. Now, if we want to graph this using transformations, we would want something that looks like the following. We would want y equals. Well, let me back up for a second. What's our basic graph? The basic graph in play here, since it's a quadratic function is going to be our basic graph is going to be y equals x squared, which you can see that on the screen right here. That's just the basic parabola of which we're going to transform. But the types of transformations we can do here, we're going to get our what the so-called vertex form we had seen previously for quadratic functions. The vertex form looks like y equals a, which that a will incorporate a vertical stretch or a vertical reflection x minus h squared plus k, which x minus h will do a vertical shift plus k will do a, excuse me, x minus h will do a horizontal shift plus k will do a vertical shift like so. And so we don't have to worry about horizontal stretching compressions because power functions are made out of Play-Doh. And so this is the general thing we have right here, but that's not what this thing looks like. How do we get to this vertex form? Well, with a quadratic function, we can always complete the square. If you can complete the square, then what this will do is it'll put your quadratic function into vertex form for which the transformations are more readily available to see. So g of x, if we're going to complete the square, I'm going to separate the variables x from the constant. I'm just going to use a basically parentheses to act like a curtain. You're going to factor out the leading coefficient from the x squared and from the x. So you factor out the negative three that leaves behind an x squared and a minus 2x. You'll notice I'm leaving a gap right here for our guest of honor. The guest of honor is identified by the following. Take our number here, we're going to take half of that linear coefficient, which half of two is equal to one. We're then going to square that number, which gives us a one. And that's who we add inside that number. So we added a one inside, we have to then subtract one, you know, so that one plus one minus one cancels out. But as this negative three would distribute on all three of these pieces, we need to make sure we subtract one times negative three. And so then computing that we have a negative three times x squared minus 2x plus one. You'll notice that x square minus 2x plus one because we complete the square. This is a perfect square trinomial. It actually would factor as x minus one quantity squared. And then we have negative one plus three, which is going to be a plus two, which we see right here, the vertex form of this parabola, for which now in the vertex form, we can see the vertical and horizontal transformations of the graph that we couldn't see before. So our vertical transformations, what do we have going on there? You're going to have a reflection across the x-axis. You have a stretch by three and a shift up by two. So let's see, we're going to reflect across the x-axis. We're going to vertically stretch the graph by three. And then there's going to be a shift up by two. Those are our vertical transformations. Our horizontal transformations, the only things we have to worry about there, are going to be this negative two right here. That's a shift to the right, a shift to the right by one. And so that helps us identify the vertex of the parabola. You know, we shifted it up by two to the right by one. So the new vertex will be one one, like so. And then we can apply the transformation. It should be concave downward because it got reflected across the y-axis. And then it should be, it's more stretched out, stretched by a factor of three compared to this one. So it looks skinnier because we elongated it. We stretched it out by a factor of three. So now getting back to the heart of the question, how does this affect domain and range? Well, for your standard function, y equals x squared, in that situation, the domain is all real numbers, and its range is going to be zero to infinity. So what happens by the transformations? So if we take our function g right here, what's the domain of g? Well, the domain of g is only going to, it's going to be the original domain of x squared, but we apply any transformations which are horizontal, which if you add one to negative infinity and infinity, you just get back negative infinity infinity. The domain doesn't change there. That's, that's typical for quadratic functions. On the other hand, if we consider the range of g, vertical transformations can change that. So the original range, right, was zero to infinity. If we reflect that across the x-axis, we times everything by negative one, that's going to give us negative infinity to zero. Infinity times negative one is negative infinity, zero times negative one is still zero. Then if we stretch everything by a factor of three, we'll times these end points by three. So we get negative three times infinity, that's still negative infinity. Zero times three is still zero. And then we shift everything up by two. We're going to add two to these numbers. And now we get our final result. Okay, negative infinity plus two is still negative infinity, and zero plus two would then be two. So the range of this function is going to be negative infinity to two, which looking back on the graph again, we see exactly that. We can get everything below two going down towards negative infinity, but nothing above. Let's look at one more example of this. Let's consider the function h of x equals 10x plus 11 over 2x plus c. This is a linear fraction of this. It's a linear function divided by a linear function. It turns out that every linear function, every linear fraction, excuse me, can be written as transformations of the following basic function. The basic graph here we're going to use is the reciprocal function y equals one over x. Now in order to see the transformations, we can't see it in this improper fraction form. We have to do some polynomial division. You could do long division, you could do synthetic division or something like that. Notice if you do synthetic division, your numerator is 10x plus 11. If you're going to do synthetic division, it's like, oh, this isn't quite work. You could factor out the two and get something like x plus one, but then you have to remember the times by the two again at the end. So I'm a little bit hesitant to do it, but let's just try it anyways. I think, let me show you what I'm having in mind here. So this first thing here, because this is just to avoid long division, which is really not too long, but if you factor out one half out from the bottom, you're going to get 10x plus 11 over x plus one, like so. Now if we do synthetic division, because we're going to divide by negative one, you can use the coefficients 10 and 11, like so. Bring down the 10. 10 times negative one is negative 10 plus 11 is one, like so. So this tells us that h of x can be written as one half times. Your quotient here is going to be 10 plus one is your remainder over the denominator, which is x plus one. And so if you distribute the one half through, what we've now discovered is that h of x can be written as one over two times x plus one plus five. So this now starts to look like a transformation of the graph, y equals one over x, like so. You can see the transformations. Now I'm actually going to rewrite the first part a little bit differently, because the way I had it written, it was involving a horizontal compression. I actually want to think of it as a vertical stretch. So I'm actually going to write it as one half times one over x plus one, like so. And so now we're in a position where we can write our transformations. What transformations are going into play here? So we're going to do a vertical compression. We're going to do a vertical compression by a factor of two. So we're times everything by one half. We're going to do a shift up by five. We see that right here. And then we're going to do a shift right by, excuse me, not, excuse me, we're going to do a shift left by one, because you see that plus one that's going on right there. So these are the transformations in play to the original graph like that. So how does that affect the graph? Well, the standard one over x has an asymptote at the y, the x-axis and the y-axis. What do these transformations do to them? Right? If you have the x-axis, for example, as that asymptote, you vertically stretch it. Well, the coordinate zero doesn't change. If you, if you shift it by five, that does move it up here. And if you shift it to the left, well, it's still just a line horizontal. This will move the horizontal asymptote to be y equals five. If you take the vertical asymptote, what, which is the y-axis, if you vertically stretch it, it doesn't do anything. If you shift it up by five, that doesn't change a vertical line. But if you shift it over to the left by one, that does. And so we see that the asymptotes of this shifted graph will be y equals five and x equals one. We can see those things. Okay? And then moving the rest of the graph, it got shrunk by a factor of one half. So it's going to look a little bit more squished compared to the standard one, what you see right there. And so then you see the graph of h of x in play right here. How does this affect the domain and the range? Well, the original graph, right? Bringing it over here, the original graph, its domain is going to be everything except for zero. And its range is going to be everything except for zero. That's the original graph. So we know the domain and the range of the basic graph. How does this affect the transformations? What is going to be the domain of h? Well, when you look at horizontal transformations, the only thing you did is you shifted everything over by one. So that, that whole at zero, then gets moved over by one. And so the domain is going to be negative infinity to negative one union, negative one to infinity, for which if we were looking for, if we were looking for problems in the domain, we would have gravitated towards that pretty quickly. Oh, what makes the denominator go to zero? Two x plus two equals zero, which means two x equals negative two, which means x equals negative one. We found that point. We could have found that without the transformations. The range is a little bit harder, right? How is the range affected by these transformations? Well, horizontal transformations don't affect the range. If you take the, if you take the original range, negative infinities to zero, union zero to infinity, if you times each of those things by one half, what happens? Well, negative infinity times one half isn't, is negative infinity. Zero times one half is still zero and zero times infinity is still infinity. So stretching it didn't change the range. If you shift everything up by five, how does that affect things? Negative infinity plus five is still negative infinity. Zero plus five, that changes. That gives you five. Zero plus five is five. And then infinity plus five is still infinity. So we actually get that the range is going to be everything except for five, which is the location of the, which is the location of the horizontal asymptote. So when it comes to linear fractionals, we can, we can do an approach like this using transformations, but in a nutshell, it always turns out to be this simple. The domain is going to be everything except for the location of the vertical asymptote. The range of a linear fractional is going to be everything, all real numbers, except for the location of the horizontal asymptote, for which the horizontal asymptote we can find, like we did before. You just look at the ratio of leading terms, right? You're going to get y equals 10 over two, which is equal to five. I wanted to show you this, this approach using transformations. So for linear fractals, we can find their range and domain using much simpler terms.