Examples finding the domain of functions

In this video, we're going to do a few examples finding domains of functions. So, let's say that we have the function f of x is equal to x plus 5 over x minus 2. What is going to be the domain of this function? Pause this video and try to figure that out.

All right, now let's do it together. Now, the domain is the set of all x values that, if we input it into this function, we're going to get a legitimate output. We're going to get a legitimate f of x. And so, what's a situation where we would not get a legitimate f of x? Well, if we input an x value that makes this denominator equal to zero, then we're going to divide by zero, and that's going to be undefined.

And so we could say the domain, the domain here is all real values of x such that x minus 2 does not equal 0. Now, typically, people would not want to just see that such that x minus 2 does not equal 0. And so we can simplify this a little bit so that we just have an x on the left-hand side. So if we add 2 to both sides of this, we would get… actually let me just do that.

Let me add 2 to both sides. So, x minus 2 not equaling 0 is the same thing as x not equaling 2. And you could have done that in your head as well. If you wanted to keep x minus 2 from being 0, x just can't be equal to 2. And so, typically, people would say that the domain here is all real values of x such that x does not equal 2.

Let's do another example. Let's say that we're told that g of x is equal to the principal root of x minus seven. What's the domain in this situation? What's the domain of g of x? Pause the video and try to figure that out.

Well, we could say the domain, the domain is going to be all real values of x such that… are we going to have to put any constraints on this? Well, when does a principal root function break down? Well, if we tried to find the principal root, the square root of a negative number, well, that would then break down.

And so, x minus seven, whatever we have under the radical here, needs to be greater than or equal to zero. So such that x minus seven needs to be greater than or equal to zero. Now, another way to say that, if we add 7 to both sides of that, that would be saying that x needs to be greater than or equal to 7. So let me just write it that way.

So such that x is greater than or equal to 7. So all I did is I said, all right, where could this thing break down? Well, if I get x values where this thing is negative, we're in trouble. So, x needs to be greater… x minus 7, whatever we have under this radical, needs to be greater than or equal to zero. And so, if you say that x minus seven needs to be greater than or equal to zero, you add seven to both sides, you get x needs to be greater than or equal to positive seven.

Let's do one last example. Let's say we're told that h of x is equal to x minus 5 squared. What's the domain here? So let me write this down. The domain is all real values of x.

Now, are we going to have to constrain this a little bit? Well, is there anything that would cause this to not evaluate to a defined value? Well, we can square any value. You give me any real number, and if I square it, I'm just going to get another real number.

And so x minus 5 can be equal to anything, and so x can be equal to anything. So here, the domain is all real values of x. We didn't have to constrain it in any way like we did the other two. The other two, when you deal with something in a denominator that could be equal to zero, then you got to make sure that that doesn't happen because that would get you an undefined value.

Similarly, for a radical, you can't take the square root of a negative, and so we would once again have to constrain on that.