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### Course: Algebra 1 (Illustrative Mathematics) > Unit 8

Lesson 3: Domain & range- What is the domain of a function?
- What is the range of a function?
- Worked example: domain and range from graph
- Domain and range from graph
- Determining whether values are in domain of function
- Identifying values in the domain
- Examples finding the domain of functions
- Determine the domain of functions
- Worked example: determining domain word problem (real numbers)
- Worked example: determining domain word problem (positive integers)
- Worked example: determining domain word problem (all integers)
- Function domain word problems

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# Worked example: determining domain word problem (real numbers)

Determining the domain of a function that models the height of a plant over time.

## Want to join the conversation?

- ok I'm totally lost. I have no idea how to get the interval of the domain and I've watched these videos multiple times.(9 votes)
- Look at the example in the video.

1) It tells you the function is called "h(t)". This tells you that the input value is the variable "t". So, your domain will be based upon the values for "t".

2) The problem tells you that "t" is the days from the time she bought the plant. This tells you that "t=0" would be the time she bought the plant. This gives you a reference point to work from.

3) The plant started growing 2 days**before**she bought it. This tells you that "t" can = -2 (remember, t=0 is when she bought the plant). This is your lowest value for "t".

4) The problem tells you she had the plant for 98 days after buying it. So, "t" can go up to 98. This is the largest value that "t" can become.

5) Thus, domain = {-2, 98] and since "t" can be in fractions, you use real number.

Hope this helps.(50 votes)

- Why integers over real numbers? Well, if the plant height was only measured once a day, integers seems like it would make more sense. We don't have enough information though.

Also, plants have height even after they die. I think the end point for the interval of the domain should be the day it was discarded.(17 votes)- This is obviously a simplification of reality. A real-world modeling function would much more advanced, requiring skills you've not studied yet.

Plants stop growing when they die, so there is no longer any need to continue modeling their height after that day: whether it is the same, a week or a month after they die, we can just just their final height.(23 votes)

- How can you decide whether it is an integer or real number? I thought an integer for days would make more sense.(11 votes)
- Integers are whole numbers, while real numbers can be any number. Real numbers would make more sense in this case, since you can track the progress of the plant between each whole day.(17 votes)

- Did anyone else notice that
*beautiful*was spelled wrong in the video?(13 votes) - What's the difference between a real number and an integer?(8 votes)
- What is the difference between parentheses () and brackets [](6 votes)
- The question says, "t days from the times she bought it" and so how can the domain include -2, which the time she didn't own the plant yet? Why doesn't it start at 0, from the time she bought it? Also, what are integers and real numbers and why is real number the better choice for the domain?(5 votes)
- The phrase "t days from the time she bought it" would mean that t=0 is when she bought it. It doesn't specify that "t" can't be less than 0.

The 1st sentence in the problem tells us that the plant spouted (so started growing) 2 days before she bought it. This is where the t = -2 comes from because the plant started growing 2 days before she bought it.

Integers are numbers like: -5, 0 3, -11, 14, etc. There are no fractions and no decimals.

Real numbers include integers, but also include all fractions and decimals.

So, why is the domain real numbers? Because plants grow in fractional units.

Hope this helps.(5 votes)

- I don't think real numbers are appropriate for this. Because real numbers also have irrational numbers. Eg. 1.333 recurring days doesn't make sense right! So I think integers is more correct. You could also have kept rational numbers instead of real numbers.(2 votes)
- 1.3333 repeating = 1 1/3. You absolutely can have 1 1/3 days. That would be 1 day and 8 hours.

Time is continuous just like a number line is continuous. Our ability to measue time may be limited by the precision of our devices. But, that doesn't mean we can't make more precise devices to get to smaller slices of time. So, real numbers does make sense.(8 votes)

- If the domain set is {-2,28}, why couldn't Sal have selected integers, as the choice to represent the domain? What makes Real numbers the best choice?(3 votes)
- He can't select integers because time does not move in increments of 1 day. Time is continuous. We measure time in days, hours, minutes, seconds and fractions of seconds.

So, the domain must be in real numbers.(6 votes)

- Shouldn't the interval be (0, 98] since the function specifically measures the height in accordance with the days 'since the time she bough it' & not 'since the time it started growing'?(3 votes)

## Video transcript

Pooja had a beautiful plant. The plant
began sprouting 2 days before Pooja bought it, and she had it for 98 days
before it died. At its tallest, the plant was 30
centimetres tall. Alright, let h of t denote the height
of Pooja's plant h, measured in centimeters, t days from
the time she bought it. Which number type is more appropriate
for the domain of the function? They tell us whether it's integers or real numbers.
So we just need to remind ourselves. The domain of a function, that's a
set of all the inputs for which the function is defined. And the inputs here, this is t and
it represents days and if I input t, the number of days,
into the function h, it'll output the height of the plant. So let's
think about it. At first, you might say, in terms of, 0 days, 1 days,
2 days, and you might be tempted to say integers. But why not think about one and a half
days or 3.175 days? So I don't see any real -- not any reason
why t couldn't be a subset of real numbers. Why you can think about, you know, the 90 -- 97.99 day. So I would say real numbers. And let's see.
Define the interval of the domain. So t. Let's see, the t would be defined up to 2 days
before she bought the plant. So I would say that t is equal to negative 2, all the way to
98 days. And t could be equal to negative 2. That would be
2 days before she bought it, or as high as 98. So let's think about it.
So the interval -- I would include, I would include negative 2.
I would include the low point, so that's why I'm
using the bracket. If I wanted t to be greater than, not greater than or equal
to, I would do a parentheses. But since t could be equal to negative 2, I'm gonna use the brackets. And at the high end, it's 98, and we're going to include 98. So I'm
gonna put the brackets there as well. So t would be a member of
real numbers such that it is a member of this
-- of this interval right over here. So negative t-- t could
be negative 2 but -- right I would say negative 2 is going to be less than
or equal to t, which is going to be less than or equal to 98. Let's check our answer. Got it right.