What are a reasonable domain and range of the function f(x) = (x - 200) * 0.15 for sending x text messages?
Understand the Problem
The question is asking to identify the reasonable domain and range for the function that describes the additional cost of sending text messages, given that the first 200 messages are free and subsequent messages incur a cost.
Answer
The domain is $[0, \infty)$ and the range is $[0, \infty)$.
Answer for screen readers
The reasonable domain of the function is $[0, \infty)$, and the reasonable range is $[0, \infty)$.
Steps to Solve
- Identify the domain of the function
The domain represents the possible values for $x$, the number of text messages sent. Since the first 200 messages are free, $x$ must be at least 0, so $x \geq 0$. Additionally, since you can't send a negative number of messages, the lower limit is $x = 0$. There is no upper limit specified for the number of messages, so we can say that $x$ can be any value greater than or equal to 0. Thus, the domain is $[0, \infty)$.
- Understand when the cost begins
For $x \leq 200$, the cost is $0$, meaning $f(x) = 0$. The cost starts at $x = 201$. Therefore, the function begins contributing to the cost at $x = 201$.
- Determine the range of the function
To find the range, we consider the values of $f(x)$ when $x > 200$. The formula for additional cost is given by $f(x) = (x - 200) \cdot 0.15$.
For $x = 201$: $$ f(201) = (201 - 200) \cdot 0.15 = 1 \cdot 0.15 = 0.15 $$
As $x$ increases indefinitely, $f(x)$ also increases. Thus, the range starts at $0.15$ and continues to positive infinity. Therefore, the range is $[0, \infty)$.
The reasonable domain of the function is $[0, \infty)$, and the reasonable range is $[0, \infty)$.
More Information
This function illustrates a common scenario in cell phone plans, where initial usage is free, and charges only apply after reaching a certain threshold. Understanding this helps consumers assess their potential costs based on their usage.
Tips
- Ignoring the initial free messages: Sometimes, students might overlook the fact that the first 200 messages do not incur a charge, leading to incorrect assumptions about the domain and range.
- Not considering limits properly: Students may give a limited range when in fact the cost continues to escalate as more messages are sent.
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