Question

What are three different representations of the domain of a function given by the number of hours slept in one day?

Answer 1

The domain of a function describes the possible input values for that function. In the case of a function representing the number of hours slept in one day, we can represent the domain in three different ways:

1. **Set Notation**: The domain can be represented as a set of possible values. For example, if we consider that a person can sleep anywhere from 0 hours (not sleeping at all) to a maximum of 24 hours (which is not realistically sustainable for most people), we can write the domain as:
\[
D = \{ x \in \mathbb{R} \mid 0 \leq x \leq 24 \}
\]
Alternatively, if considering typical sleep patterns, we might limit the upper bound to a more realistic maximum sleep duration, such as:
\[
D = \{ x \in \mathbb{R} \mid 0 \leq x \leq 12 \}
\]

2. **Interval Notation**: The domain can also be expressed using interval notation. For the realistic sleep scenario assuming a maximum of 12 hours, the domain could be expressed as:
\[
D = [0, 12]
\]
Or, if we consider the theoretical maximum of 24 hours, it would be:
\[
D = [0, 24]
\]

3. **Graphical Representation**: The domain can be visually represented on a number line. You would draw a line from 0 to 24 (or 12), shading the area between these two points to indicate that all values between the endpoints are included. For example, a number line with points labeled from 0 to 24, with a solid line or shading covering that interval, would visually represent the domain:
```
------------|----------------------------|------------
0 24
```
Alternatively, if considering a more typical sleep duration:
```
------------|------------------|-------------
0 12
```

These different representations highlight the range of hours that can be realistically expected for sleep in a day.