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### Course: Get ready for AP® Calculus > Unit 4

Lesson 3: Interpreting function notation# Function notation word problem: bank

Learn how to interpret expressions that contain a function within a real-world context. In this video, the function we interpret models an account balance over time.

## Want to join the conversation?

- Why is the initial amount defined as M(0) and not M(1)?(31 votes)
- no this is wrong because m=the amount of money in the account and (t) is the days and on this one it says (t) it the money from when he opened it.(6 votes)

- How do you determine if a function is exactly a function?(23 votes)
- You can use the vertical line test with graphs. Essentially, for every x-value, there should not be more than 1 y-value. When just given an equation, the f(x) notation shows that its a function. Also, if you input an x-value into the equation of a function, you should get only one answer (only one y-value).(3 votes)

- maybe math isnt my thing..(12 votes)
- Try changing the way you understand the information. I personally have a hard time listening to the videos; I just look at the diagrams. Or try to take a day off because stress isn’t the most helpful thing in the world.(6 votes)

- I'm so sorry.... I've lost sleep over this.

I believe the answer is the 1st choice, not the 3rd choice as Sal stated.

1. It would take an initial deposit of $120,000 to make close to $100 interest in the first month assuming 1% average annual int-rate on savings account which would be .08333% per month.

2. The function refers to the account balance after 30 days, not the interest earned (or in this case "profit"). The 3rd choice states he earned a "profit" of $100 after 30 days which is what Sal picked.

I think it more likely that the account was very close to the opening balance of $100 after 30 days given the pitiful interest paid. Besides, who the heck invests $120,000 in an account with a return rate of 1% annually?

What's the error in my thinking? :-)(6 votes)- The problem asks you to figure out what M(30)-M(0)=100 represents.

M(30) = the balance in the account after 30 days.

M(0) = the balance in the account after 0 days (or basically that starting balance.

Thus, M(30)-M(0) = Current balance after 30 days - Initial balance (amount invested) = 100

The 100 has to be the profit because you have subtracted the intial amount invest, the M(0).

Hope this helps.(15 votes)

- What happens if you don’t understand it and u watched it like 10 times.

😕(8 votes)- You can try other sources...(7 votes)

- tomorrow is another day(10 votes)
- What is initial balance?(5 votes)
- Instead of calling it as initial balance, lets call that as the principal amount we invested in something.

So, M(30) is the final amount we have after thirty days.

So, when we subtracted the principal amount we invested from the final amount we had at the end of thirty days, we made $100 dollar profit.

i.e.,**M(30)-M(0)=$100**

Hope it helps :)(5 votes)

- what is function notation? we weren’t even taught what it is and now we’re doing word problems?(3 votes)
- Go to the start of this unit. It is discussed in the 1st video and the others before you get to this one. Here's a link to the beginning.

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:evaluating-functions/v/what-is-a-function(4 votes)

- if he just opened up the bank account how can he have a profit of $100 if not depositing any money?(3 votes)
- The problem description tells you in the first line that the account was opened with an initial sum. It doesn't tell you how much. You just need to assume that it was enough that after 30 days, there was a profit of $100.(2 votes)

- can you save this kind of money in real life like this.(3 votes)
- yes, it is possible.(2 votes)

## Video transcript

Arjun opened up a savings account last
year and put an initial sum in it. Let M of t
denote the account balance M measured in dollars t days since it was opened. What does
the statement M of 30 minus M of 0 equalling a hundred mean?
Before I look at the choices, let's think about what this means. When you input t equals 30 into your
function, you're going to get M of 30.
So let me make that clear. So if you say t is equal to 30, you input that into your
function M. You're going to get -- you're going to
get M of 30. So one way to think about it
is -- This is the account balance 30 days
since it was opened. This is when t is equal to 30. This is the account balance
after 30 days Let's write that down. Balance -- balance after 30 days. Now, by the same logic, this
right over here this is when we put t where we said t equals 0. This is the balance
after 0 days, or you could say this is the initial balance. initial -- initial balance. So what they're doing -- they're taking our
balance after 30 days, and from that they're subtracting the initial
balance and they're saying that equal to 100. So there's a couple ways you can
interpret this and I haven't even looked at these choices yet. We'll see if
any of these match up. You could say that your balance after 30
days is a hundred dollars more than the initial balance. Or another way to think about is
you added a hundred dollars in the first 30 days. Those are both legitimate ways to think
about it. I'll see which of these choices are consistent with that. 30 days after it was
opened, the balance of Arjun's account was equal to 100. No, that's not what that's
saying. This statement right over here -- the
balance 30 days after opening -- this statement right over here -- this
would be equivalent. This is equivalent to saying that, M of 30 -- this is the balance after 30
days after it was opened -- is equal to 100. That's not what they tell us here.
They tell us that the difference between the balance after 30 days and
the initial balance -- that's a hundred. So we can rule that one out. Arjan had
the initial amount of money in his account 30 days after he opened it. So if he had
the same amount -- if he had the initial amount -- let me write this down. So had the initial
amount of money -- the initial amount of money is M of 0. So they're saying he had the
initial amount of money in his account 30 days after he opened it. Well, the amount
that he had in his account 30 days after he opened it is M of 30. So these are the same amounts of money
then this -- In order to be consistent with this,
you would have an equation like this. M of 0, the initial amount, is equal to the amount after 30 days.
That's not what they told us over here. We can rule that out. And then finally we have the choice Arjan made a profit
over a hundred dollars over the first 30 days since the account
was opened. That seems reasonable that his balance
is a hundred dollars higher. The difference between -- if you take the initial
balance and subtract it from his balance after 30 days, it's a hundred. And this right over here is a
hundred higher than his initial balance. So it makes sense that maybe he got the
profit out of an interest or something else that he got in his bank account over the
first 30 days.