Welcome back, everyone. So, up to this point, we've talked about the 3 main types of transformations: 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 when 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. 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. 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, 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 12 units from the position 0. By looking at this graph, if I go ahead and try to 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.

To really solidify this concept, let's try an example to see how we do. 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) = (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. Here, the h corresponds to this 3 because we have x - h within the function, and inside the square function, we have x - 3. Our h is 3. Also, our k value, which is positive 2, modifies the original position. Since the h was positive, the graph will shift right and since the k is positive, the graph will shift up. So our new parabola is going to go 3 units to the right and 2 units up.

If I look at the domain and range of this new parabola, I can see that the domain is going to be all real numbers because this parabola expands in all directions to the left and right. So, our domain goes from negative infinity to positive infinity for this new function g(x). But what about the range? Well, originally, the range of the initial parabola goes from 0 to infinity because we can see that on the y-axis, it goes from 0 and continuously goes up. But after our transformation, our range goes from 2 to infinity, meaning our range is from 2 to infinity and includes this value. This demonstrates how the domain stays the same, but the range changes when we shift 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.