Hey, everyone, and welcome back. So in the previous video, we talked about how we could evaluate composed functions. Now in this video, we're going to be taking a look at finding the domain of functions which have been composed. Now this process can often be a bit tricky because it can be difficult to recognize how the domain of the functions will behave when they're separate versus how the domain acts when the functions are composed together. But don't worry about it because in this video, we're going to be going over an example that I think will really clear this entire concept up. So without further ado, let's get right into this.

Now when finding the domain of composed functions, there are a few steps you can follow, and I think it's best to see these steps by looking at an example. So given below is an example where we have 2 functions. We have f(x)=1x-2 and g(x)=x. We're asked to determine the composition f∘g(x) and find its domain. Now keep in mind that this f∘g(x) that we see here is the same thing as f∘g(x), where g is the inside function.

Now your first step with dealing with these types of problems is to find the x values not defined for g(x). So what you first want to do is find any restrictions on the inside function. Now if I look at our g function, we can see here that g(x), the inside function, is equal to x. Now recall that we've talked about in previous videos how the square root of x does have restrictions, because we cannot have a negative number underneath the square root. You cannot take the square root of something negative. So in this case, we would say that x has to be greater than or equal to 0 because x cannot be negative, so this would be the restrictions for our function g.

Now once you've found these restrictions, your next step should be to find any x values that make g(x) not defined for f(x). And if you're not sure what this means, basically, what this is saying is if we take g and we put it inside of f, what restrictions are we going to have on that entire function when we compose the two together? So to figure this out, what I'm going to do is actually compose these functions. So we'll have f∘g(x). Now in doing this step, what I can do is take our function g and I can place it inside of the function f. So I'll just replace this x with the function we have here. So rather than having one over x minus 2, our function f∘g(x) is going to be 1x-2, because the square root of x is this function g.

Now we already determined the restrictions for the square root of x, which is that x has to be positive, but notice that when we plug g into f, we get this new fraction that we see here, this new kind of equation, and when we do this, we see that for a fraction, the denominator cannot equal 0. That's another restriction that we have. So we can say that this whole denominator, x-2, cannot be equal to 0. Now what I can do from here is solve for x in this mini equation by adding 2 on both sides; that'll get the positive and negative 2 to cancel, giving me that the square root of x cannot equal 2. From here, I can take both sides of this equation and square it. This will allow me to cancel the square and the square root, giving me that x cannot equal 22, which is 4. So this right here is the restriction for our function f∘g(x).

And notice at this point, we have found the restriction for g, which was the first step, and we found the restriction for f∘g(x), which was the second step. Since we have found these two restrictions, we can now write out what our domain is. So the domain for our composed function f∘g(x) is that we can go from 0, and we are including the 0 because x can be greater than or equal to 0, so we can go from 0 all the way up to 4, not including 4, and then we can go from 4 to infinity. So basically, we could have any number as long as it's above 0 and not equal to 4. So this right here would be the domain of our function f∘g(x). And notice how we found first the domain for g, then we found f∘g(x) domain, and then we combine these together to get the total domain. So this is how you can find the domain of composed functions. Hopefully, you found this video helpful, and thanks for watching.