Math  /  Algebra

QuestionConsider the function B(t)B(t) that gives the temperature of Boston, Massachusetts on a particular day, where tt is measured in hours after midnight. Is B(t)B(t) a one-to-one function? How do you know?

Studdy Solution

STEP 1

What is this asking? Is there only *one* time for each temperature in Boston on this day, or could the same temperature happen at different times? Watch out! Just because the temperature goes up and down doesn't automatically mean it's one-to-one!

STEP 2

1. Think about temperature
2. Relate to one-to-one functions

STEP 3

Let's imagine a typical day in Boston.
The temperature usually starts out cool in the early morning, then warms up during the day, reaching a peak sometime in the afternoon.
Then, as evening approaches, the temperature starts to drop again.

STEP 4

So, what does this tell us?
It means the temperature probably *isn't* always increasing or always decreasing throughout the entire day.

STEP 5

Remember that a **one-to-one function**, sometimes called an **injective function**, means that for every output, there's only *one* input that produces it.
In our case, this would mean that for every temperature, there's only one time of day when it occurs.

STEP 6

But, as we just discussed, the temperature in Boston likely goes up, reaches a peak, and then comes back down.
This means that the same temperature could occur at *two different times* during the day!
For example, the temperature at 88 am might be 60F\text{60}^\circ \text{F}, and then later, at 88 pm, the temperature might be 60F\text{60}^\circ \text{F} again.

STEP 7

In math terms, we could say there exist t1t_1 and t2t_2 such that t1t2t_1 \ne t_2 but B(t1)=B(t2)B(t_1) = B(t_2).

STEP 8

B(t)B(t) is *not* a one-to-one function because the same temperature can occur at multiple times throughout the day.

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