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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.

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  • leaf blue style avatar for user Scott Trask
    Why is the initial amount defined as M(0) and not M(1)?
    (31 votes)
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  • starky sapling style avatar for user Javiel Velez
    How do you determine if a function is exactly a function?
    (25 votes)
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    • piceratops ultimate style avatar for user Genius71752
      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)
  • aqualine seed style avatar for user Susan Sizemore-LaPorte
    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? :-)
    (5 votes)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (6 votes)
  • hopper jumping style avatar for user Ellie Kim
    What is initial balance?
    (5 votes)
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    • duskpin sapling style avatar for user Nitish
      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 :)
      (3 votes)
  • piceratops seed style avatar for user bloodclot
    maybe math isnt my thing..
    (5 votes)
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    • duskpin tree style avatar for user Chloe Han
      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.
      (2 votes)
  • leafers ultimate style avatar for user Mamirov Yakhyo
    If f(x+2)=x^3+6x^2+12x+8, what is f(x^1/3)? could you help, please
    (2 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      In this problem, we just have to work backwards. If we are given f(x + 2), all we have to do to find f(x) is substitute x with (x - 2), since that will bring us back to x from x + 2. Adding the +2 essentially moved the graph 2 units to the left, so we have to move the graph two units to the right to get f(x), which is by inputting (x - 2) for x:
      f(x + 2 - 2) = (x - 2)^3 + 6 (x - 2)^2 + 12 (x - 2) + 8
      f(x) = (x^3 - 6x^2 + 12x - 8) + 6 (x^2 - 4x + 4) + 12x - 24 + 8
      f(x) = x^3 - 6x^2 + 6x^2 + 12x - 24x + 12x - 8 + 24 - 24 + 8
      f(x) = x^3 + 0x^2 + 0x +0
      As you can see, this all simplifies to just f(x) = x^3. Now we simply take the 1/3rd power of this, which is the same as taking the cube root. Taking the cube root of a simply yields the number that was cubed in the beginning. Thus:
      f(x^1/3) = (x^1/3)^3
      f(x^1/3) = x ^ (3/3)
      f(x^1/3) = x ^ 1 = x
      Hope this helps!
      (3 votes)
  • duskpin ultimate style avatar for user Mai N.
    I don't get why the 2nd option doesn't work for the equation? Doesn't the 2nd option state that the account open for 30 days?
    (2 votes)
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  • leafers ultimate style avatar for user anika
    What happens if you don’t understand it and u watched it like 10 times.
    😕
    (2 votes)
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  • duskpin ultimate style avatar for user 雨辞生
    tomorrow is another day
    (2 votes)
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  • blobby green style avatar for user smallbinary
    Doesn't M(t) = M EXACTLY? (They did take the trouble of stating to us "account balance M..." as part of our data. ) If so, it seems there's some willed deception here at not trumpeting it. It "gets us to think," but isn't the goal getting us to compute? Why make it trickier?

    By the way, any f(x) is always vertical? Any a(b), b(c), d(e), f(g)...z(k) asf - all these vertical?
    (2 votes)
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    • duskpin ultimate style avatar for user astra1
      If you are graphing a function, usually the input will be on the x-direction and the output will be on the y-direction. In M(t) t is the input, M is the function and M(t) is the output given t, but you can replace them with any variable, and the definitions will not change.
      (1 vote)

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.