A coral mobility cell phone plan costs 75$/month and gives you unlimited talk and 1000 free text messages for months. After 1000 text messages, they charge $0.04 per text. What is the domain and range of the function?

Let's define a function to represent the cost of the Coral Mobility cell phone plan, based on the number of text messages sent per month.

### Function Definition:
- Let $ T $ represent the number of text messages sent in a month.
- The cost $ C(T) $ is $75 for the first 1000 text messages.
- For any text messages beyond 1000, there is an additional charge of $0.04 per text.

Thus, the cost function $ C(T) $ can be expressed as:

\[
C(T) =
\begin{cases}
75, & \text{if } T \leq 1000 \\
75 + 0.04(T - 1000), & \text{if } T > 1000
\end{cases}
\]

### Domain of the Function:
The domain refers to all possible values of $ T $, the number of text messages.

- Since $ T $ cannot be negative (you can't send a negative number of texts), the domain is all non-negative integers:
\[
\text{Domain} = [0, \infty) \quad \text{(or } T \geq 0\text{)}
\]

### Range of the Function:
The range refers to the possible values of the cost $ C(T) $.

- For $ T \leq 1000 $, the cost is a constant $75.
- For $ T > 1000 $, the cost increases linearly by $0.04 for every additional text message.

Thus, the range is:
\[
\text{Range} = [75, \infty) \quad \text{(as the cost starts at $75 and can increase indefinitely depending on the number of texts sent)}
\]

### Summary:
- **Domain:** $ [0, \infty) $
- **Range:** $ [75, \infty) $

Would you like further clarification on how we derive this? Let me know if you have any questions!

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**Related Questions:**
1. How does the cost increase as the number of texts goes beyond 1000?
2. What would the cost be if you sent 1500 text messages in a month?
3. Can the cost function be represented graphically? How would it look?
4. Is there a similar pricing model for data usage in cell phone plans?
5. How do functions with piecewise definitions behave in general?

**Tip:** In piecewise functions, the domain for each part must be defined clearly to avoid confusion, especially at transition points (like 1000 texts here).

Understand the domain and range of a piecewise function representing the Coral Mobility cell phone plan costs. The plan offers unlimited talk and 1000 free text messages for $75/month, with an additional $0.04 charge per text message beyond 1000. Explore the function's domain as non-negative integers and the cost range starting from $75.

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