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## Determining the domain of a function

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# Examples finding the domain of functions

## Video transcript

- [Instructor] In this video, we're gonna 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 five over x minus two. 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 that the domain, the domain here is all real values of x, such that x minus two does not equal zero. Now typically, people
would not want to just see that such that x minus
two does not equal zero, 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 two to both sides of this, we would get, actually,
let me just do that. Let me add two to both sides. So x minus two not equaling zero is the same thing as x not equaling two, and you could have done
that in your head, as well. If you wanted to keep x
minus two from being zero, x just can't be equal to two, and so typically, people
would say that the domain here is all real values of x such that x does not equal two. Let's do another example. Let's say that we're told that g of x is equal to the principle
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 that 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 principle
root function break down? Well if we tried to
find the principle 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 is if we add seven to both sides of that, that would be saying that
x needs to be greater than or equal to seven, so let me just write it that way. So such that x is greater than or equal to seven. 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 seven, whatever we have in this, under the 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 five 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. To give me any real
number and if I square it, I'm gonna just get another real number, and so x minus five 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 the denominator that
could be equal to zero, then you've got to make
sure that doesn't happen 'cause that would get
you an undefined value and 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.