To determine if the given table represents a function, we need to understand what defines a function in mathematical terms. A function is a relationship between two sets, typically called the domain (the set of inputs, often denoted as [tex]\( x \)[/tex]) and the range (the set of outputs, often denoted as [tex]\( y \)[/tex]), where each input [tex]\( x \)[/tex] is related to exactly one output [tex]\( y \)[/tex].
Let’s analyze the table:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Hours of Training} (x) & \text{Monthly Pay} (y) \\
\hline
10 & 1250 \\
\hline
20 & 1400 \\
\hline
30 & 1550 \\
\hline
40 & 1700 \\
\hline
50 & 1850 \\
\hline
60 & 2000 \\
\hline
70 & 2150 \\
\hline
\end{array}
\][/tex]
We need to check if each [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value:
- For [tex]\( x = 10 \)[/tex], [tex]\( y = 1250 \)[/tex]
- For [tex]\( x = 20 \)[/tex], [tex]\( y = 1400 \)[/tex]
- For [tex]\( x = 30 \)[/tex], [tex]\( y = 1550 \)[/tex]
- For [tex]\( x = 40 \)[/tex], [tex]\( y = 1700 \)[/tex]
- For [tex]\( x = 50 \)[/tex], [tex]\( y = 1850 \)[/tex]
- For [tex]\( x = 60 \)[/tex], [tex]\( y = 2000 \)[/tex]
- For [tex]\( x = 70 \)[/tex], [tex]\( y = 2150 \)[/tex]
From this analysis, it is evident that each distinct [tex]\( x \)[/tex]-value (hours of training) maps to exactly one unique [tex]\( y \)[/tex]-value (monthly pay).
According to the options given, the correct explanation is:
A. Yes, because every [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value.
Thus, the table indeed represents a function because for each input [tex]\( x \)[/tex] in the domain, there is a unique output [tex]\( y \)[/tex] in the range.
