What is the domain of D(m)?
Understand the Problem
The question is asking for the mathematical domain of the function D(m), which represents the distance walked in miles after m minutes. Given that they walk a 1-mile route that takes 20 minutes, the possible values of m should fall within the range of 0 to 20 minutes.
Answer
The domain of $D(m)$ is all real numbers from 0 to 20.
Answer for screen readers
The domain of $D(m)$ is all real numbers from 0 to 20.
Steps to Solve
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Understanding the Function's Input The function $D(m)$ represents the distance walked in miles after $m$ minutes.
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Identifying the Range of Time Since Quincy takes 20 minutes to walk 1 mile, $m$ can only take values from 0 (start of the walk) to 20 (end of the walk). Thus, the interval is $0 \leq m \leq 20$.
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Defining the Domain The domain of $D(m)$ is the set of all valid input values for which the function is defined. In this case, it includes all real numbers between 0 and 20, inclusive.
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Formulating the Domain Hence, the domain of $D(m)$ can be expressed as all real numbers from 0 to 20, which is written as $[0, 20]$.
The domain of $D(m)$ is all real numbers from 0 to 20.
More Information
The function $D(m)$ is constrained because Quincy walks a fixed route in a specific amount of time. Therefore, the realistic limits on $m$ ensure that it cannot take on any values outside of this defined interval.
Tips
- Assuming all real numbers: Many may mistakenly think that $m$ can be any real number, but that’s incorrect due to the context of the problem.
- Confusing whole numbers with real numbers: It's important to distinguish between whole numbers and real numbers, as the timing can be expressed in decimals.
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