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Domain of advanced functions

Sal covers many different kinds of functions and shows how to determine their domain. Created by Sal Khan.

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  • blobby green style avatar for user ram.the.great.625
    HOW COME 1/O IS WRITTEN AS UN-DEFINED OR INFINITE? CAN SOMEONE PLZ EXPLAIN HOW 1/O IS UNDEFINED?????????????????
    (0 votes)
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    • leafers ultimate style avatar for user GFauxPas
      Jennifer is correct, but here's another way to show that it's impossible to divide 1 by 0.
      Let's say that there is an answer. Let's call that answer "q"
      1/0 = q
      Let's multiply both sides by zero
      1 = 0*q
      1 = 0
      Ridiculous!

      The answer can't be infinity either because infinity isn't a number and "defined" means the answer is a number.
      (305 votes)
  • starky ultimate style avatar for user Luke
    At Sal says that f(x)=x^2 has a domain of all real numbers. Sqrt(-1) or i is not a real number, but when plugged into f(x) produces a valid output. What's going on here?
    (8 votes)
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    • piceratops ultimate style avatar for user Just Keith
      While it is necessary to constrain the domain of the function to avoid numbers that give undefined outputs, there is nothing that says one cannot further constrain the domain for other reasons.

      For example, suppose you measured velocities and distances only after the first hour of travel. Your resulting equations might make mention that time has to be greater than or equal to 1 hour. That doesn't mean that the math of the equations wouldn't give you a number for t = 0.50 hours, but that your equation itself wasn't set up to include that time range, so the output (while a single number) might be incorrect.

      So, when a domain is constrained, that only means you didn't intend for that equation to be used for the forbidden values -- for any number of reasons.
      (7 votes)
  • blobby green style avatar for user Hetav45
    In the last function, should not the domain be Z(set of integers) - {1} as f(x) is only defined for even and odd numbers?
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      I would tend to agree with you. I looked on google to try and determine if decimals or fractions are ever considered even or odd. There is some disagreement. But, most sites seem to indicate the concept of even and odd numbers only applies to integers. I think your domain would be better than Sal's.
      (5 votes)
  • duskpin ultimate style avatar for user Το Αίμα του Υιού της δημιουργίας και της καταστροφής, ο Υιός του Triscalade
    well if 1/0 is defined, dividing by zero or nothing, basically not dividing at all would just still be one wouldn't it?
    (4 votes)
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  • purple pi purple style avatar for user Judith Jones
    At 1.37 It seems to me that the domain would include some imaginary numbers too, since i squared gives an answer of -1 and 3i squared gives an answer of -9. Am I wrong? Or do you just only think about real numbers when stating the domain?
    (2 votes)
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    • leaf red style avatar for user Age of Caffeine
      That's why we need to state the domain. In normal exercises, the domain must be a subset of the real numbers ((so no imaginary numbers)), but as you say, complex numbers are completely fine; they are just not expected as answers for "domain questions".

      Now, many times the domain including complex numbers can make a huge difference, as when you say "but that cannot be true as the domain of the function f is the real numbers" at the end of a reduction to absurd proof. We could say that the domain is a "variable" part of a function, just as its definition, so for example
      f(x) = sin x ((0 < x < pi / 2))
      g(x) = sin x ((x belonging to Complex numbers))
      are completely different functions, as their domains are different.
      (7 votes)
  • blobby green style avatar for user Mychelly Moraes
    Shouldnt be only X > 3? Because if X < -3 wouldnt be possible to have 0 as result?
    (0 votes)
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  • male robot hal style avatar for user pnorris9030
    Sal says he forgets if colon or a line is used. Which is it?
    (3 votes)
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    • male robot hal style avatar for user Jesse
      Actually, you could use either. Some texts use a colon, and others use a vertical bar. Other texts use a backward rounded epsilon, which I have always found confusing. So how you annotate "such that" is up to you (or perhaps your teacher).
      (2 votes)
  • piceratops ultimate style avatar for user Far_Lost
    THIS QUESTION IS OFF TOPIC, DUE TO THE FACT THAT I AM UNABLE TO FIND INTERVAL NOTATION

    A question that has been confusing me for some time now :

    *If you were given a function (ex. -10 < X <= 10, for simplicity reasons), and asked for interval notation, how would you do so correctly?*

    Using the example formula below, here's how I went about the problem :
    *(The higher line represents the thick line you graph on paper)*
    <----◙⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃|⁃⁃⁃⁃○---->
    -10 -6 -2 0 2 6 10

    After doing so, the interval notation I believed it to be is : `[10, 10)

    According to my teacher, I was wrong. IS she right?
    (2 votes)
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    • marcimus pink style avatar for user Tyler
      First, here's the section of Khan Academy on interval notation:

      https://www.khanacademy.org/math/algebra/algebra-functions/domain-and-range/v/introduction-to-interval-notation

      Now then, on to your question.

      -10 < X <= 10
      In interval notation, that would be:
      (-10, 10]

      The reason for the parentheses on the left is because X cannot equal "-10", since the original notation says that "-10 < X". When X cannot equal that thing, you use parentheses instead of brackets.

      The reason for the bracket on the right is because X can equal 10. When X can equal that number, you use brackets.

      As for the number line you drew, that would appear to be:
      [-10, 10)
      OR
      [-6, 10)

      it seems a little off, because spacing doesn't work so well on Khan Academy posts. So I can't tell whether the -10 or the -6 was supposed to be under that black circle on the left of the number line.

      I should point out, however, that the number line and the example you gave above (-10 < X <= 10) are not the same thing. The number line is (-10 <= X < 10), not the other way around.

      Anyways, I hope this helps. And if I misunderstood any part of your question, just comment on my answer and let me know.
      (3 votes)
  • orange juice squid orange style avatar for user Nick Timms
    Would 0/0 still be undefined?
    (1 vote)
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  • blobby green style avatar for user LC
    Lets say a plumber costs C (t) = 35 + 25t dollars per hour for jobs up to four hours long, where t is the number of hours. Find the values C(t+0.5) -C(t)? Then figure out for what values of t your answer will be true. Note that domain of the function is {t:0≤t≤4}.

    Any ideas? How can I answer a question in this form, it isn't a homework question..I'm just trying to understand how to solve it myself! Thanks in advance!
    (1 vote)
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Video transcript

Welcome to my presentation on domain of a function. So what's is the domain? The domain of a function, you'll often hear it combined with domain and range. But the domain of a function is just what values can I put into a function and get a valid output. So let's start with something examples. Let's say I had f of x is equal to, let's say, x squared. So let me ask you a question. What values of x can I put in here so I get a valid answer for x squared? Well, I can really put anything in here, any real number. So here I'll say that the domain is the set of x's such that x is a member of the real numbers. So this is just a fancy way of saying that OK, this r with this kind of double backbone here, that just means real numbers, and I think you're familiar with real numbers now. That's pretty much every number outside of the complex numbers. And if you don't know what complex numbers are, that's fine. You probably won't need to know it right now. The real numbers are every number that most people are familiar with, including irrational numbers, including transcendental numbers, including fractions -- every number is a real number. So the domain here is x -- x just has to be a member of the real numbers. And this little backwards looking e or something, this just means x is a member of the real numbers. So let's do another one in a slight variation. So let's say I had f of x is equal to 1 over x squared. So is this same thing now? Can I still put any x value in here and get a reasonable answer? Well what's f of 0? f of zero is equal to 1 over 0. And what's 1 over 0? I don't know what it is, so this is undefined. No one ever took the trouble to define what 1 over 0 should be. And they probably didn't do, so some people probably thought about what should be, but they probably couldn't find out with a good definition for 1 over 0 that's consistent with the rest of mathematics. So 1 over 0 stays undefined. So f of 0 is undefined. So we can't put 0 in and get a valid answer for f of 0. So here we say the domain is equal to -- do little brackets, that shows kind of the set of what x's apply. That's those little curly brackets, I didn't draw it that well. x is a member of the real numbers still, such that x does not equal 0. So here I just made a slight variation on what I had before. Before we said when f of x is equal to x squared that x is just any real number. Now we're saying that x is any real number except for 0. This is just a fancy way of saying it, and then these curly brackets just mean a set. Let's do a couple more ones. Let's say f of x is equal to the square root of x minus 3. So up here we said, well this function isn't defined when we get a 0 in the denominator. But what's interesting about this function? Can we take a square root of a negative number? Well until we learn about imaginary and complex numbers we can't. So here we say well, any x is valid here except for the x's that make this expression under the radical sign negative. So we have to say that x minus 3 has to be greater than or equal to 0, right, because you could have the square to 0, that's fine, it's just 0. So x minus 3 has to be greater than or equal to 0, so x has to be greater than or equal to 3. So here our domain is x is a member of the real numbers, such that x is greater than or equal to 3. Let's do a slightly more difficult one. What if I said f of x is equal to the square root of the absolute value of x minus 3. So now it's getting a little bit more complicated. Well just like this time around, this expression of the radical still has to be greater than or equal to 0. So you can just say that the absolute value of x minus 3 is greater than or equal to 0. So we have the absolute value of x has to be greater than or equal to 3. And if order for the absolute value of something to be greater than or equal to something, then that means that x has to be less than or equal to negative 3, or x has to be greater than or equal to 3. It makes sense because x can't be negative 2, right? Because negative 2 has an absolute value less than 3. So x has to be less than negative 3. It has to be further in the negative direction than negative 3, or it has to be further in the positive direction than positive 3. So, once again, x has to be less than negative 3 or x has to be greater than 3, so we have our domain. So we have it as x is a member of the reals -- I always forget. Is that the line? I forget, it's either a colon or a line. I'm rusty, it's been years since I've done this kind of stuff. But anyway, I think you get the point. It could be any real number here, as long as x is less than negative 3, less than or equal to negative 3, or x is greater than or equal to 3. Let me ask a question now. What if instead of this it was -- that was the denominator, this is all a separate problem up here. So now we have 1 over the square root of the absolute value of x minus 3. So now how does this change the situation? So not only does this expression in the denominator, not only does this have to be greater than or equal to 0, can it be 0 anymore? Well no, because then you would get the square root of 0, which is 0 and you would get a 0 in the denominator. So it's kind of like this problem plus this problem combined. So now when you have 1 over the square root of the absolute value of x minus 3, now it's no longer greater than or equal to 0, it's just a greater than 0, right? it's just greater than 0. Because we can't have a 0 here in the denominator. So if it's greater than 0, then we just say greater than 3. And essentially just get rid of the equal signs right here. Let me erase it properly. It's a slightly different color, but maybe you won't notice. So there you go. Actually, we should do another example since we have time. Let me erase this. OK. Now let's say that f of x is equal to 2, if x is even, and 1 over x minus 2 times x minus 1, if x is odd. So what's the domain here? What is a valid x I can put in here. So immediately we have two clauses. If x is even we use this clause, so f of 4 -- well, that's just equal to 2 because we used this clause here. But this clause applies when x is odd. Just like we did in the last example, what are the situations where this kind of breaks down? Well, when the denominator is 0. Well the denominator is 0 when x is equal to 2, or x is equal to 1, right? But this clause only applies when x is odd. So x is equal to 2 won't apply to this clause. So only x is equal to 1 would apply to this clause. So the domain is x is a member of the reals, such that x does not equal 1. I think that's all the time I have for now. Have fun practicing these domain problems.