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What is the domain of a function?

Functions assign outputs to inputs. The domain of a function is the set of all possible inputs for the function. For example, the domain of f(x)=x² is all real numbers, and the domain of g(x)=1/x is all real numbers except for x=0. We can also define special functions whose domains are more limited.

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  • female robot amelia style avatar for user 4Abbyrose1
    I really didn't get the x€R thing. It didn't make any sense. We already covered this in school and it never came up. What is it?
    (100 votes)
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    • leaf blue style avatar for user Stefen
      The ϵ is the Greek letter epsilon. In this context epsilon is a symbolic way of saying "member of", so when you see "x ϵ" it means "x is a member of."

      Suppose you were a part of a cupcake club called "The Yummies". Then 4Abbyrose1 ϵ The Yummies would read 4Abbyrose1 is a member of The Yummies. Now while you wouldn't use ϵ when you are writing English, you use it a lot when writing math - and there are many many more symbols that are a kind of "math Shorthand".

      Often you will see something like x ϵ R, which means "x is a member of the real numbers", or you can just say "x is a real number". If you saw n ϵ Z, that means n is a member of the integers, or you can just say n is an integer.

      Remember when you used "x" as the multiplication sign? - Well, ϵ may seem weird now, but later, as you become more used to it, you will appreciate it. If you continue in math, you'll meet it often.
      (294 votes)
  • leafers sapling style avatar for user themayankgoel28
    Sal says that g(y) = (y-6)^1/2.
    Let y be 15.
    (15-6)^1/2
    (9)^1/2
    or +3 or -3. But as per definition can a function have 2 values for a given input?
    (23 votes)
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  • blobby green style avatar for user Randall Pennington
    If division by zero is undefined,then wouldn't the square root of zero also be undefined??In your example you give yis greater than or "equal" to six.If y=6 then you end up with the squared roo of zero,don't you??
    (10 votes)
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    • aqualine tree style avatar for user Judith Gibson
      Division by zero IS NOT defined because there's NO number that when multiplied by zero gives the original (non-zero) number that's being divided by zero.
      But there is ONLY ONE number that when multiplied by itself gives the answer of zero --- and that is zero. So the square root of zero IS defined and is zero.
      (36 votes)
  • duskpin seed style avatar for user Jazlin Ramirez
    I'm still confused, how do you know if the domain or range is a real number?
    (8 votes)
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  • piceratops ultimate style avatar for user Ashley O'brien
    For the second example to the video, at approximately , I'm wondering why negatives aren't possible with the square root sign. I can see how it has to be greater than zero, its only the greater than six that confuses me.
    (8 votes)
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    • piceratops ultimate style avatar for user Waxillium Ladrian
      the value of y has to be greater than or equal to 6, or else the number under the radical would become negative. As we know, the square root of a negative number is not real, thus y cannot be less than 6.

      For eg, for y = 5, g(y) = (-1)^1/2, which is not a real number. Therefore, y >= 6.
      (6 votes)
  • leaf blue style avatar for user wooyoung jung
    What are the definitions of real numbers?
    (5 votes)
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  • blobby green style avatar for user Merlyn Davies
    Is the domain always the x-value
    (5 votes)
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  • blobby green style avatar for user Adriana Dyson
    This concept makes sense to me when Sal does it with smaller numbers but I'm having trouble with a specific homework problem and I'm not sure how to use what Sal teaches in this video to go about solving it...

    The question is as follows:

    g(x)= x+1/x^2-2x-15

    (function of x equals 1 plus x all over x squared minus 2x minus 15)

    Any help or tips would be super appreciated!!
    (2 votes)
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    • starky ultimate style avatar for user KLaudano
      g(x) = (x + 1) / (x^2 - 2x - 15)

      The function will be defined for all real numbers except when the denominator equals 0. (We cannot divide by 0 after all.) The easiest way to determine when the denominator equals 0 is to factor the quadratic equation.

      g(x) = (x + 1) / ((x - 5) * (x + 3))

      As you can see, the denominator will be 0 when x = -3 or x = 5. So, the domain of g is all real numbers except -3 and 5.
      (10 votes)
  • blobby green style avatar for user alanis peguero
    G (y) =√ (y-6) has a sqr so it going to have 2 outputs, does it still being a function??
    (5 votes)
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  • spunky sam green style avatar for user Vijay
    Why did Sal describe the function g(y) in terms of real numbers? I mean that we do know about complex numbers and the imaginary number i, which is useful for finding square roots of negative numbers. Why didn't he simply consider y belonging to the set of Complex numbers? Was it to make the function look easier and to make it understandable by other people?
    (4 votes)
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Video transcript

Let's have a little bit of a review of what a function is before we talk about what it means that what the domain of a function means. So function we can view as something -- so I put a function in this box here and it takes inputs, and for a given input, it's going to produce an output which we call f of x. So, for example, let's say that we have the function -- let's say we have the function f of x is equal to 2 over x. So in this case if -- let me see -- that's my function f. If I were to input the number 3. Well, f of 3 that we're going to output -- we have, we know how to figure that out. We've defined it right over here. It's going to be equal to 2 over 3. It's going to be equal to 2 over 3. So we're able, for that input, we're able to find an output. If our input was pi, then we input into our function and then f of pi -- when x is pi, we're going to output f of pi, which is equal to 2 over pi. So we could write this as 2 over pi. We're able to find the output pretty easily. But I want to do something interesting. Let's attempt to input 0 into the function. If we input 0 then the function tells us what we need to output. Does this definition tell us what we need to output? So if I attempt to put x equal 0, then this definition would say f of 0 be 2 over 0, but 2 over 0 is undefined. Rewrite this -- 2 over 0. This is undefined. This function definition does not tell us what to actually do with 0. It gives us an undefined answer. So this function is not defined here. It gives a question mark. So this gets to the essence of what domain is. Domain is the set of all inputs over which the function is defined. So the domain of this function f would be all real numbers except for x equals 0. So we write down these, these big ideas. This is the domain -- the domain of a function -- Actually let me write that out. The domain of a function A domain of a function is the set of all inputs -- inputs over which the function is defined -- over which the function is defined, or the function has defined outputs over which the function has defined outputs. So the domain for this f in particular -- so the domain for this one -- if I want to say its domain, I could say, look, it's going to be the set of these curly brackets. These are kind of typical mathy set notation. I said OK , it could be the set of -- I gonna put curly brackets like that. Well, x can be a member So this little symbol means a member of the real numbers. But it can't be any real number. It could be most of the real numbers except it cannot be 0 because we don't know -- this definition is undefined when you put the input as 0 So x is a member of the real numbers, and we write real numbers -- we write it with this kind of double stroke right over here. That's the set of all real numbers such that -- we have to put the exception. 0 is not a -- x equals to 0 is not a member of that domain -- such that x does not -- does not equal 0. Now let's make this a little bit more concrete by do some more examples So more examples we do, hopefully the clearer this will become. So let's say we have another function. Just be clear, we don't always have to use f's and x's. We could say, let's say we have g of y is equal to the square root of y minus 6. So what is the domain here? What is the set of all inputs over which this function g is defined? So here we are in putting a y it to function g and we're gonna output g of y. Well it's going to be defined as long as whatever we have under the radical right over here is non-negative. If this becomes a negative, our traditional principal root operator here is not defined. We need something that -- if this was a negative number, how would you take the principal root of a negative number? We just think this is kind of the the traditional principal root operator. So y minus 6, y minus 6 needs to be greater than or equal to 0, in order for, in order for g to be defined for that input y. Or you could say add six to both sides. y is to be greater than or equal 6. Or you could say g is defined for any inputs y that are greater than or equal to 6. So you could say the domain here, we could say the domain here is the set of all y's that are members of the real numbers such that y, such that they're also greater than or equal, such that they're also greater than or equal to 6. So hopefully this is starting to make some sense -- You're all used to a function that is defined this way. You could even see functions that are divided fairly exotic ways. You could see a function -- let me say h of x -- h of x could be defined as -- it literally could be defined as, well h of x is gonna be 1 if x is equal to pi and it's equal to 0 if, if, x is equal to 3. Now what's the domain here? And I encourage you to pause the video and think about it. Well, this function is actually only defined for two input. If you, we know h of -- we know h of pi -- if you input pi into it we know you're gonna output 1, and we know that if you input 3 into it h of 3, when x equals 3, you're going to -- you're going to -- put some commas here. You're gonna get 0. But if you input anything else, what's h of 4 going to be? Well, it hasn't defined. It's undefined. What's h of negative 1 going to be? It hasn't defined. So the domain, the domain here, the domain of h is literally -- it's just literally going to be the the two valid inputs that x can be are 3 and pi. These are the only valid inputs. These are the only two numbers over which this function is actually defined. So this hopefully starts to give you a flavor of why we care about to the domain. It's not all functions are defined over all real numbers. Some are defined for only a small subset of real numbers, or for some other thing, or only whole numbers, or natural numbers, or positive numbers, and negative numbers. So they have exceptions. So we'll see that as we do more and more examples.