Welcome back everyone. So up to this point, we've talked about the three main types of transformations being reflections, shifts and stretches and shrinks. Now, in this video, we're going to take a look at how we can find the domain and range of a function. After it's been transformed, it's called that you're going to see scenarios where you have to find the domain and range of something where a transformation has acted on it. So it's important that we know how to solve these types of problems when we come across them. So let's get right into this. A transformation can change the domain and range of a function. Now, when finding the domain and range of a function that has been transformed, you can actually do this by observing whatever the new graph looks like after the transformation. Now, we talked about in previous videos how to find the domain and range of a graph. But just as a refresher, let's try finding the domain and range of this function F of X. So to find the domain, we can imagine taking our graph and squishing it down to the X axis if we were to squish this graph down, it would look like a line that goes from negative three to positive three. So this tells us our domain. Now, if we want to find the range of this graph, we can imagine taking this graph and squishing it down to the Y axis. If we squish the graph down to the Y, we're going to end up with a line on the Y axis that goes from negative three to positive three as well. And that's our range. So pretty straightforward for finding the domain and range of this graph. But what if we had a transformation that acted on this function? Would we have the same domain and range? Well, we discussed that this could change the domain and range. So let's see what happens. Notice we have the same overall shape but it's been shifted to a new location. Specifically, we've been shifted to 12, the position 12 from the position 00. So let's see what happens here. Well, by looking at this graph, if I go ahead and try and squish this thing down to the X axis, I'm going to get a line that goes from negative two to positive four. So our domain goes from negative 2 to 4. And if I want to find the range of this graph, I can squish this down to the y axis which will give me a range from negative one all the way up to positive five. So our range is go from negative 1 to 5 and notice how the domain and range that we got are different than the, than the domain and range we had in the original function. So this just goes to show you that a transformation can change our domain and range. Now to really solidify this concept. Let's try an example to see how we do. So here we're given a function F of X is equal to X squared. And we're asked to sketch a graph of the function G of X is equal to X minus three squared plus two and determine its domain and range. Now the function X squared is just going to be a parabola centered at the origin. And if I look at the transformation that we're given, I noticed that this looks to be in the form F of X minus H plus K which is a shift transformation. Now, I can see here that the H corresponds with this three right here because we have X minus H within the function and then inside the square function, we have X minus three. So I can tell that our H is going to be three. Now, I can also see that our K value is going to correspond to this positive two. So our K is positive two, since the H was positive, the graph will shift to the right. And since the K is positive, the graph will shift up. So our new parabola is going to go 123 units to the right and 12 units up, meaning we're going to be at this point right here. So notice we have the same Parabola, but it's been shifted to a new location, specifically, it's been shifted to 32. So this is the new position. Now, if I look at the domain and range of the Parabola, I can see here that the domain is going to be all real numbers because notice how this Parabola just expands in all directions to the left and right. So there's going to be no restrictions on our domain. So we can say our domain goes from negative infinity to positive infinity for this new function that we got G of X. But what about the range? Well, originally we have the, our range of the initial problem goes from zero to infinity because we can see here that on the y axis it goes from zero and continuously goes up. But after our transformation, notice that we went from this range to a range that goes like this. So our range is really going to go from positive two, all the way to infinity, meaning our range is going to go from two. And we need to include this value to infinity and this would be the range of our graph. So notice how the domain stayed the same but the range was different when we shifted our graph. So this is how transformations can change the domain and range of your function. Hopefully, you found this video helpful. Let me know if you have any questions.