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Worked example: determining domain word problem (all integers)

Determining the domain of a function that models going up and down a ladder.

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  • leaf red style avatar for user Hannah
    Why can't the interval of the domain be [0,15]?
    (13 votes)
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    • stelly blue style avatar for user Kim Seidel
      On word problems, read, reread, and reread again. it is easy to miss little details. In the 1st sentence, it tells us that Mason is standing on the 5th step. Then, in the end of the 2nd paragraph it tells us that if Mason moves down, then "n" will be a negative number. The combination of these 2 pieces of info tell us that the 5th step = 0. Since Mason can go down, the interval can't start at 0. It needs to start at the lowest number that "n" can represent. This would be the ground, where n = -5. Thus, the interval is [-5, 10].
      Hope this helps.
      (43 votes)
  • aqualine ultimate style avatar for user HAHA
    So what's the difference between integer and real numbers?
    (14 votes)
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    • aqualine ultimate style avatar for user Jack Messina
      Integers are just whole numbers, 1, 2, 3, 4, 5....-1, -2, and so on. Integers encompass all numbers that exist. Real numbers are crazy decimals, fractions, and whole numbers. You can think about integers as falling into the category of real numbers. Does that help?
      (24 votes)
  • female robot grace style avatar for user Suhana S.
    I'm still kind of confused on how to find the interval and I have watched all videos about it. Could someone explain it to me?
    (4 votes)
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    • blobby green style avatar for user isaacfink21
      The interval of the domain is a range of all the possible inputs that work in a function. For example, if you walk to a hotdog stand containing 30 hotdogs that cost $1 each, your domain is how many possible inputs (hotdogs you can buy) work. In this situation, the "interval" or "range of domains" is from 0 to 30. You can't buy negative hotdogs so you cant go under 0, and you can't buy more than 30 hotdogs because that's all thats in stock.

      This also works on a graph where the "x" value is your input and your "y" value is your output. This means every point's x-value on the graph is a possible input and has a corresponding output which is it's y-value. To find the range of domains, or interval of possible inputs, we must look at the range of valid x-values (because the x-values are the valid inputs). This works so long as the graph is not infinite in the x-direction. To find the interval, just look at how far the x-value reaches on each side, or the most negative input and most positive input. You can also find the interval of outputs, or the "range" by finding how far up and down in the y-direction the graph goes.

      Hope this helps!
      (12 votes)
  • starky seedling style avatar for user nameless
    How is 0.5m pronounced? O.5 metres or 0.5 metre?
    (5 votes)
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    • mr pink green style avatar for user David Severin
      This is an oddity in math. If you say 0.5, you would say point five meters, but if you have 1/2, you could say either one half meters or more often you may here one half of a meter. meter (singular) is used with one meter, and in the second case of 1/2, it is used as a unit ratio.
      (7 votes)
  • mr pink red style avatar for user Maybe Curtis
    At , since the domain of the solution is all integers, is the range of the function be all real number because there is 0.5 meter?
    (4 votes)
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    • stelly blue style avatar for user Kim Seidel
      The domain was not all integers as the ladder does not extend forever. The domain is only integers from -5 up to 10 inclusive.
      Yes, the range would need to be real numbers, but not all real numbers. Again, it will be bounded started at h(-5) = 0 up to h(10) = 7.5. A more accurate representation of range would be to create a set of specific values. Since the steps increase by 0.5 meters, the range can't really be all real numbers from 0.5 to 7.5. A more accurate range is the set = {0, 0.5; 1.0; 1.5; ... 6.5; 7.0; 7.5}
      (7 votes)
  • purple pi purple style avatar for user Narayani Pokhrel
    I am sorry for having to bring this up here, I am probably not supposed to, but in the 'function domain word problems' exercises, I really don't understand the way this problem -> -> (Fernanda is addicted to the game "Candy Birds," where she has to eat as many candies as possible without getting hit by mischievous birds.)<- <- is solved in the exercise.

    If anyone can please go to the exercise and get to this question and understand what is happening in the hints especially, Please please please make me understand.
    (6 votes)
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  • blobby green style avatar for user dcronin05

    In the video Sal indicates that steps are an integer because you can't have a part of one. But in one of the practice problems the input variable is a unit of time, specifically minutes. The hints explain that time is continuous so the domain should be real numbers. This however seems subjective when compared to the ladder problem as minutes are also a unit that isn't generally described in fractions as there are other units used (seconds) when describing parts of a minute. I get that this could be considered semantics but I'm curious if i'm missing something or if there is a hard rule in these situations or if it really matters at all and the concept is more important?
    (4 votes)
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    • stelly blue style avatar for user Kim Seidel
      When you go up/down a ladder, you move one step at a time. There are no 1/2 steps or 1/4 steps. You can only move one step up or down. This is why it is only defined for integers.

      Time is measured in years, months, days, hours, minutes, seconds and fractions of seconds. The units can be very small. They are only limited by the precision of the device used to measure time. Time does not have to be in minutes. If something takes 1.5 minutes, we can and do use the fractional values.

      Hope this helps you to see the difference.
      (3 votes)
  • blobby green style avatar for user je259109
    The function definition would be f(n) = 0.5n + 2.5
    (5 votes)
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  • piceratops sapling style avatar for user Anusha Jain
    at , why does he Sal say root 2 steps or pi steps when both aren't real nos. ?
    (1 vote)
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  • aqualine ultimate style avatar for user cconley
    im confusion. if 2/0 is undefined, then how is 0/2 0. it just doesnt follow logic in my opinion but whatever plz help me now
    (0 votes)
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    • stelly blue style avatar for user Kim Seidel
      If the problem was 8/2 you would find the number that multiplied by 2 creates 8. The answer is 4 because 4x2 = 8.

      Now, apply this same approach to 0/2. What number times 2 creates 0? The answer is 0 because 2x0 = 0

      Next try 2/0. What number times 0 creates 2? This one is impossible. Any number times 0 = 0. There is no way to create the 2.

      Hope this helps.
      (8 votes)

Video transcript

- [Voiceover] This right over here is a screenshot from a Khan Academy exercise, and it says, "Mason stands on the 5th step of a vertical ladder. "The ladder has 15 steps, and the height difference "between consecutive steps is 0.5 meters. "He is thinking about moving up, down, or staying put." Let me draw this ladder that Mason is on. It's a vertical ladder, that's one side of the ladder, this is the other side of the ladder, and it has 15 steps. Let me see if I can draw that. This is the first one, two, three, four. I'm gonna run out of space, I need to make 'em closer together. It's gonna be one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, and 15. 15 steps. Let me make sure it's even. The top and the bottom, and the distance between each of these, I guess you could say, steps, or the rungs of the ladder, are half a meter. This distance right over here is 0.5 meters. And it says that he's on the 5th step of this vertical ladder. He's on the 5th step, One, two, three, four, five. This is where he is right now. He's on this 5th step. And he's thinking about moving up, or down, or staying put. "Let h of n denote the height above the ground h "of Mason's feet (measured in meters) "after moving n steps (if Mason went "down the ladder, n is negative.)" All right, h of n. Denote the height above the ground after moving n steps. Make sure we understand this. If I were to say h of zero, what is that going to be? Well, h of zero means that he's moved zero steps. He's moved zero steps, he's still going to be on this 5th step of the ladder. And so how high is he going to be? If he's on the 5th step of a vertical ladder... I'm assuming that there's 0.5 from the ground. This is the ground right over here. He is one, two, three, four, five steps, each of 'em is half a meter. Five times 0.5 is going to give us 2.5 meters, so h of zero is 2.5 meters. If I said h of one, that means he goes up. H of one means he goes up one step. Here, n would be equal to one. If he goes up one step, h of one, he's going to half a meter higher, so it's going to be equal to three meters. We could keep doing that for a bunch of different inputs. Let me write that, that's going to be equal to three meters. But anyway, that's not what they're asking us about. They're saying, "Which number type is more appropriate "for the domain of the function?" The domain, just as reviewed, that's the set of numbers that we could input into the function and get a valid output. And it's clear here, see, we have to pick between integers or real numbers. Well, n, which is our input, that's the number of steps he goes up or down. It could be positive or negative, but we're not gonna talk about half steps. Then he'll put his foot in the air, right over. He has to take integer valued steps up or down. Or, I guess, he's taking integer valued steps, if it's positive it's up, if it's negative it's down, if it's zero that means he's staying put. If n is zero, that means he's staying put. It's not real numbers. He can't move pi steps from where he is. He can't move square root of two steps from where he is. He can't even move 0.25 steps, then he'd put his foot in the air. This is definitely going to be about the integers, not the real numbers. This function right over here, the valid inputs, I want to be able to input an integer. In fact, it's not even all integers, because he can't go down an arbitrary amount. In fact, he can't go up an arbitrary amount either. The domain is going to be a subset of integers. Then they say, "Define the interval of the domain." And we have these little toggles here... to define the interval of the domain. And let's see, the lowest value for n, he can go as far as one, two, three, four, five steps down. In that case, n would be equal to negative five. And then the highest value for n is if he takes one, two, three, four, five, six, seven, eight, nine, 10 steps up. And so that would be n is equal to 10. The interval of the domain, and actually I just copy and pasted this onto my scratch pad, n can be as low as negative five, and as high as 10, and it can include them as well, so I'm gonna use brackets. My domain would include negative five. If it didn't include negative five, I would put a parentheses, but I could put brackets here, and I could put brackets there as well. Just for fun, let me actually input it into the actual exercise. I'm saying integers, and I'm as low, and I can go down five steps, and I can go up 10 steps, and 10 is also included in my interval. Then I can check my answer, and I got it right.