Let's carefully analyze the problem and determine the constraints on the domain and range of the function [tex]\( m \)[/tex].
First, let's consider the domain of the function [tex]\( m \)[/tex], which represents the number of months, [tex]\( x \)[/tex], from the time the moss is planted. According to the given table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline$x$ & 0 & 6 & 12 & 18 \\
\hline$m(x)$ & 6 & 12 & 24 & 48 \\
\hline
\end{tabular}
\][/tex]
The domain values listed are [tex]\( 0, 6, 12, \)[/tex] and [tex]\( 18 \)[/tex]. These values describe specific points in time from the planting of the moss, and looking at these points, we can conclude that:
- The value of [tex]\( x \)[/tex] must be at least [tex]\( 0 \)[/tex] because it cannot be negative (you can't have negative months).
- The value of [tex]\( x \)[/tex] can only go up to [tex]\( 18 \)[/tex] because beyond that, we do not have any data.
Therefore, the constraint on the domain [tex]\( x \)[/tex] is:
[tex]\[ 0 \leq x \leq 18 \][/tex]
Next, let's consider the range of the function [tex]\( m \)[/tex], which denotes the area that the moss covers in square inches. According to the given table, the moss can cover areas of [tex]\( 6, 12, 24, \)[/tex] and [tex]\( 48 \)[/tex] square inches. From this, we observe:
- The minimum area that the moss covers is [tex]\( 6 \)[/tex] square inches.
- The maximum area that the moss covers is [tex]\( 48 \)[/tex] square inches.
Therefore, the constraint on the range [tex]\( m(x) \)[/tex] is:
[tex]\[ 6 \leq m(x) \leq 48 \][/tex]
With these constraints, we can summarize:
- Constraint on the domain [tex]\( x \)[/tex] is [tex]\( 0 \leq x \leq 18 \)[/tex]
- Constraint on the range [tex]\( m(x) \)[/tex] is [tex]\( 6 \leq m(x) \leq 48 \)[/tex]
Therefore, the correct answer is:
Domain: [tex]\( x \)[/tex] must be greater than or equal to 0 and less than or equal to 18.
Range: [tex]\( m(x) \)[/tex] must be greater than or equal to 6 and less than or equal to 48.
