(31-32) Consider each situation and determine if the statement represents an invertible function.
If so, write a statement that describes the inverse function. Explain why if not possible.
31. The number of miles, of a race car driver has completed in a race in terms of the time, in
hours.
32. The population, of Miami, Florida in terms of the year from 2020 to 2023 .

Ask by Byrd Harmon. in the United States

Nov 07,2024

Question

(31-32) Consider each situation and determine if the statement represents an invertible function.
If so, write a statement that describes the inverse function. Explain why if not possible.
31. The number of miles, of a race car driver has completed in a race in terms of the time, in
hours.
32. The population, of Miami, Florida in terms of the year from 2020 to 2023 .

Ask by Byrd Harmon. in the United States

Nov 07,2024

UpStudy AI · Accepted Answer

To determine if the statements represent invertible functions, we need to assess whether each function is one-to-one (bijective). A function is invertible if each output corresponds to exactly one input.

### 31. The number of miles, $$ n $$, of a race car driver has completed in a race in terms of the time, $$ t $$, in hours.

**Analysis:**
- In this case, we can assume that the race car driver is moving at a constant speed. If the speed is constant, then the relationship between time and distance is linear, represented by the equation $$ n = vt $$, where $$ v $$ is the speed.
- If the speed is constant and positive, then for every hour $$ t $$, there is a unique distance $$ n $$ that corresponds to that time. Thus, the function is one-to-one.

**Inverse Function:**
- The inverse function would express time in terms of distance: $$ t = \frac{n}{v} $$.
- This means that for a given distance $$ n $$, you can find the time $$ t $$ it took to complete that distance.

**Conclusion:**
- Yes, this function is invertible, and the inverse function can be described as $$ t = \frac{n}{v} $$.

### 32. The population, $$ p $$, of Miami, Florida in terms of the year $$ t $$ from 2020 to 2023.

**Analysis:**
- The population of a city typically changes over time, but it does not necessarily do so in a one-to-one manner. For example, the population could increase or decrease due to various factors (birth rates, death rates, migration, etc.), and it may not be strictly increasing or decreasing.
- If the population fluctuates, then there could be multiple years $$ t $$ that correspond to the same population $$ p $$. For instance, if the population were to be the same in two different years, the function would not be one-to-one.

**Conclusion:**
- No, this function is not guaranteed to be invertible because the population may not be a one-to-one function of time. Therefore, we cannot write a meaningful inverse function for this situation.
31. The function is invertible, and the inverse function is $$ t = \frac{n}{v} $$. 32. The function is not invertible due to the population's potential fluctuations over time.

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