Hi, everyone. Welcome back. In the last video, we talked about how to find the domain and range of a function based on this graph. And in this video, we're going to be looking at how you can find the domain of an equation. Now when finding the domain of an equation, this is something that is a bit more complicated than with a graph, but we are going to go over some of the common cases that you'll see that will hopefully make problems a lot more clear when you encounter them. So let's get into this.

Now recall, when finding the domain of a function, you're looking for the allowed x values that you can have, but there are going to be certain situations where you'll have restrictions on these x values. So when finding the domain of an equation, you need to first identify the values that will break the function. Any values that break your function are going to be restrictions that you'll have, and you need to be able to recognize these. There are 2 common situations where we'll have restrictions. One of the common ones is whenever you have an x inside of a square root. So the domain for x values inside a square root is going to be anything that does not make the inside of the square root negative. You do not want to see a negative number underneath a square root.

So let's take a look at this example. We're asked to find the domain of the function \( f(x) = \sqrt{x} \) without graphing and to express our answer using interval notation. Now remember, you do not want the inside of the square root to be negative. So what that means is that our x values cannot be below 0. So any x values that are below 0 will make this negative. Right? We can have \( \sqrt{1} \), that's just equal to 1. We can even have \( \sqrt{0} \) that's just equal to 0. But if we have \( \sqrt{-1} \), well, you can try plugging that into your calculator, you're going to get an error. This is not something you're allowed to have. So your x values cannot be below 0, therefore, your domain is going to be from 0 all the way to positive infinity. Now your domain can equal 0; that's perfectly fine but it just cannot be below 0. So this would be the domain of the function, and this is the answer for this example.

Now, if we actually take a look at the graph of \( \sqrt{x} \), notice that for the graph, this actually makes sense because notice that all of these values where x is above 0, these are all defined. But as soon as we look at the values where x is below 0, these values are not included. So it makes sense that our graph would be from 0 all the way to positive infinity.

Now remember, this is one of the 2 common situations that you'll see where the domain has restrictions. The other common scenario is whenever you have x in the denominator of a fraction because you need to make sure that for the domain that your x values do not make the denominator equal to 0. You cannot divide by 0 in a fraction, so this is another situation you'll commonly run into. So let's take a look at this example.

This example says given the function \( f(x) = \frac{2}{x - 5} \), find the domain using interval notation. Now to find the domain of this function, the denominator cannot be equal to 0. So I'm just going to write out that \( x - 5 \neq 0 \). And I can just solve this like a mini equation. So we'll take 5 and we'll add it to both sides, that'll get the fives to cancel, giving me \( x \neq 5 \). So that means the x values that make the denominator 0 are 5. So our domain cannot equal 5. That means that our domain is going to be every value from negative infinity all the way up to 5, but not including the 5 because we can't actually equal this number. Then we're also going to have an interval that goes from 5 all the way to positive infinity. So this basically says that we can have any real number except for the number 5, and that is the domain of the function. So that is how you can find the domain of a function if you're given an equation rather than a graph. Hopefully, you found this helpful, and let me know if you have any questions.