Give the meaningful domain and range of the exponential model

For the given exponential decay model \( N(t) = N_0 \times (1 - r)^t \), where \( N_0 = 22,000,000 \) and \( r = 0.15 \), we can define the meaningful domain and range based on the context of the problem.

### Domain

The domain of the function represents all possible values of \( t \), which in this case is the time in months. Since time cannot be negative and we are considering the decline over time, the meaningful domain is:

\[ t \in [0, \infty) \]

This indicates that the model starts at \( t = 0 \) (the initial time) and continues indefinitely into the future.

### Range

The range of the function represents all possible values of \( N(t) \), which is the number of users at time \( t \). 

- At \( t = 0 \), \( N(t) = N_0 = 22,000,000 \).
- As \( t \) increases, \( N(t) \) decreases because the application is losing users at a rate of 15% per month.
- The number of users will approach 0 as \( t \) approaches infinity but will never actually reach 0.

Thus, the meaningful range is:

\[ N(t) \in (0, 22,000,000] \]

This indicates that the number of users starts at 22,000,000 and decreases towards 0 over time.

Would you like further details or have any questions?

Here are 5 questions you might want to ask next:
1. How do you graph an exponential decay function like this one?
2. What are some real-life applications of exponential decay models?
3. How do you solve for the exact number of users after a specific time period?
4. Can exponential decay models be applied to population growth scenarios?
5. What are the differences between exponential growth and exponential decay?

**Tip:** When dealing with exponential functions, always consider the practical context to determine the meaningful domain and range. This helps in understanding the real-world implications of the model.

Explore the domain and range of exponential decay models with a focus on the formula N(t) = 22,000,000 * (1 - 0.15)^t. Understand how time affects the number of users in applications over an indefinite period. Suitable for students in Grades 10-12.

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