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# Examples finding the domain of functions

Sal shows how to algebraically find the domain of a few different functions.

## Want to join the conversation?

• Isn't the blue equation supposed to be x>7, not x≥7
• It is correct, 7 is a valid solution because √(7-7)=√0=0.
• Why does Sal say "Principal root" instead of square root?
• Square roots have two answers, so if you have sqrt(36) both 6 and -6 are correct answers. The principal root would be the positive number which creates the square root function. The bottom half of the sideways parabola (negative square roots) are not part of the function.
• the domain of squere root of x-7
khan said all real values of x such of 7 ≤ x so how can squere root of 7-7 be undefind when its defined as 0 , can someone explain it to me please
• The square root of zero is zero because zero times itself is still zero.

So the function you mentioned above is defined for all values of x that are 7 or higher.
• At , why can't x equal 2, and why does it not equal 2? I am sort of confused on how he can to that. Please Advise.
• You cannot divide by 0, so if x=2, 2-2=0, and the expression becomes indeterminate.
• What is a piecewise function?
• A function which varies for different parts of the domain, so the domain is divided into segments, and each segment could have a different function. One of common ones is stair step function with domain 0≤x<1 y=1. 1≤x<2 y=2, 2≤x<3 y=3, etc. which looks like a stair step without the vertical components.
• I have a question that involves a triangle: the base is 30 m and is the distance of a camera from a rocket launch pad. The height x increases as the camera's angle is continually adjusted to follow the base of the rocket.
I've expressed the function as: For the height x at angle theta, the relation R is R = {(theta, x) l (such that) x is the height of the base of the rocket at angle theta}.
I'm not sure if I've got that right - now I am to give the domain of the function. I think it is probably [0, ...for sure, but could it be [0, infinity)?
The question also is asking to give the height of the rocket when the elevation angle is pi/3. I have some idea on how to figure this out, but I'm at a loss. I'm taking calculus online and my tutor told be he's too busy b/c he as 160 students.
I don't know if it's ok for me to post for help on here, but I hope so.
Thank you.
• What makes a real number real? How is it different from rational/irrational numbers?
• Real numbers include rational/irrational numbers, it is just how they are grouped. What makes a real number real, is when it is not complex and also because we say it is. Another reason for we call them real, is because they work in real life. Which is a bad argument, because imaginary numbers are also useful in real life.

But if you meant real number, as in natural numbers like {1,2,3,4,5...}
If so, then they are different from irrational numbers because they aren't infinite repeating like 1/7 where it is 0.142857.... and so forth. If we want to compare a natural number to a rational one, it is a rational number. That is just because we can represent it as a natural number always by dividing by 1.
• But why isn't he taking in consideration the scenario of x=-5 in the first exercise? Because even though x≠2 it's correct, it doesn't take in consideration that x could be -5, having again an operation where there's a division by zero:

f(-5)= -5+5/-5-2 = 0/-7
(1 vote)
• There is a big difference between having the 0 in the denominator as compared to the numerator. In the numerator, 0/-7=0 so you have a value. In the denominator, 5/0 is undefined because you cannot divide by 0.
• The last function, h(x) = (x - 5)^2
x = any number?

Because I tested several different numbers from negatives to zero to positives and they all worked..
(1 vote)
• Yes, the domain of h(x) is all real numbers.
• For the domain of the equation g(x)= sqrt(x-7), can you set the domain as X≠<7?
(1 vote)
• Well, I see what you mean by that, but conventionally, it would need to be greater than 7, so the domain would be:
x > 7
Domain tells us what the x value can be. What you have is what x can't be, if that makes sense.

## 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.