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 common 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. Therefore, 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; and 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(x). 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 3 to positive 3. 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 3 to positive 3 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 position 12 from position 0. Let's see what happens here. 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 2 to positive 4, 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 5, so our range is going to go from negative one to 5, and notice how the domain and range that we got are different 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(x) is equal to x², and we're asked to sketch a graph of the function g(x) is equal to (x − 3)² + 2 and determine its domain and range. Now, the function x² is just going to be a parabola centered at the origin, and if I look at the transformation that we're given, I notice that this looks to be in the form f(x − h) + k, which is a shift transformation. Now, I can see here that the 'h' corresponds with this 3 right here because we have x − h within the function and then inside the square function, we have x − 3. So I can tell that our 'h' is going to be 3. Now I can also see that our 'k' value is going to correspond to this positive 2. So our 'k' is positive 2. 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 1, 2, 3 units to the right and 1, 2 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. Now if I look at the domain and range of this 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 are 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(x). But what about the range? Well, originally, our range on the initial parabola goes from 0 to infinity because we can see here that on the y-axis it goes from 0 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 2 all the way to infinity, meaning our range is going to go from 2 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. 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.