Interpreting function notation
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.