Hey, everyone, and welcome back. So let's see if we can solve this problem. In this problem, we have 2 functions. We have fx is equal to the square root of x plus 4 plus 30, and then we have gx is equal to the square root of x plus 4 minus 2x plus 35. We're asked to complete the following operations below and determine the domain of the new functions. So let's see if we can solve this problem. What I'm first going to do is see if I can find f + g. And to do this, recall that this is the same thing as fx plus gx. So I just need to add the 2 functions above together. I can see here that fx is equal to the square root of x plus 30. And I can see here that gx is f of x, gx is equal to the square root of x plus 4 minus 2x plus 35. Now since I don't have any kind of negative signs or numbers that I need to distribute into these parentheses over here, I can kind of just drop the parentheses on all of these functions that we plugged in here. So all I need to do really is combine like terms. Now I see that we have a square root of x plus 4 and a square root of x plus 4, which will give us 2 square roots of x plus 4. So we have 2 of these here, so we add them together. And then I also see that we have a positive 30 and a positive 35. 30 plus 35 is 65. So we're going to get plus 65, and then we would have this minus 2x. Although to do this in better order, I'm actually going to write this as minus 2x and then plus 65. Typically, we like to have the x's come before the constants. That's just a general preference. And this right here would be the functions when we added them together. So fx plus gx would look like this. Now to find the domain, what you have to do is find the combination of the domains from the functions when you initially plugged them in. Now looking at these functions when I did the initial plug-in, I can really see that the only thing that's going to give us restrictions on our domain is the square root function because it is typically the square roots or fractions that you see that will cause restrictions. And since we have a square root, we need to take this into account. Recall that nothing underneath the square root can be negative, so what we can do is take this whole x + 4 and say this has to be greater than or equal to 0. Now what I can do from here is solve this mini-equation by subtracting 4 on both sides, canceling the 4s there, giving me that x has to be greater than or equal to negative 4. So the x values can equal negative 4, but they just cannot be less than negative 4. So that means that our domain is going to go from negative 4 to positive infinity. And since I see that it's the same quantity we have under the square root, this is the only restriction that I need to take into account, meaning this is the only domain. So this right here is going to be the domain of f+g. And that's how you can solve these types of problems. But what would happen if instead we had f-g in this situation over here? Well, to solve this situation, what I'm going to do is the same strategy I used up here, except now I'm going to subtract the 2 functions. So we're going to have fx, which is the square root of x plus 4, plus 30, and this is going to be minus the gx function, which is the square root of x plus 4 minus 2x plus 35. Now from here, what I'm going to do is since I have a negative sign out front, I need to distribute this negative sign into each of these terms. So I can drop the parenthesis on this first thing; this will give me square root of x plus 4, this first quantity I should say, plus 30, and then I can take this negative sign and distribute it into each of these terms. Taking the negative sign and distributing it to here, we'll get minus square root of x plus 4. The two negative signs here will cancel giving me positive 2x, and then the negative sign multiplied by 35 will give me minus 35. So this right here is what the function is going to look like when we distribute the negative sign. Now I can see here that because we have a positive and a negative quantity for the square root of x plus 4, these are going to cancel each other out. And I can also see that we have a positive 30 and a negative 35. 30 minus 35 or 30 plus negative 35 is going to give you negative 5. So that means that what we're going to end up with is this 2x by itself, and this is going to be minus 5. So this is what the function f-gx looks like. And as for the domain, you may look at this total function and think there are no restrictions on the domain because we only have an x here and it's like we got this polynomial which is a linear equation. But if you look up here, notice that we have to take into account the functions before we simplify them. So when we initially plug things in, we have the square root of x plus 4. So because of that, the inside of the square root has to stay positive, meaning that x plus 4 still has to be greater than or equal to 0 since we started with this square root. And if I go ahead and solve this mini equation, well, we already did that up here. We already know x is going to be greater than or equal to negative 4, so that means our domain for this situation is still going to be all real numbers from negative 4 to positive infinity. So this is going to be our domain when we subtract the 2 functions. Notice it's the same for when we added them. Even though this came out to a function that didn't have the square root in it anymore, we still needed the same domain because the square root was inside the function when we initially combined them together. So this is how you can do addition and subtraction of functions and find the domain. Hope you found this video helpful, and thanks for watching.