To determine if the given table represents a function, we need to recall the definition of a function. In mathematics, a relation between two sets is called a function if every element of the first set (domain, or [tex]\( x \)[/tex]-values) is associated with exactly one element of the second set (range, or [tex]\( y \)[/tex]-values).
Given the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Hours of \\
Training
\end{tabular} &
\begin{tabular}{c}
Monthly \\
Pay
\end{tabular} \\
\hline
10 & 1250 \\
\hline
20 & 1400 \\
\hline
30 & 1550 \\
\hline
40 & 1700 \\
\hline
50 & 1850 \\
\hline
60 & 2000 \\
\hline
70 & 2150 \\
\hline
\end{tabular}
\][/tex]
Let's analyze this table step by step:
1. Examine the [tex]\( x \)[/tex]-values (Hours of Training):
- The [tex]\( x \)[/tex]-values are: 10, 20, 30, 40, 50, 60, 70.
- Each [tex]\( x \)[/tex]-value is unique; there are no repeated [tex]\( x \)[/tex]-values.
2. Examine the [tex]\( y \)[/tex]-values (Monthly Pay):
- The [tex]\( y \)[/tex]-values are: 1250, 1400, 1550, 1700, 1850, 2000, 2150.
- Each [tex]\( y \)[/tex]-value is associated with a unique [tex]\( x \)[/tex]-value.
To check if this is a function, we need to verify that each [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value. Based on our examination:
- 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]
Each [tex]\( x \)[/tex]-value has only one corresponding [tex]\( y \)[/tex]-value, and none of the [tex]\( x \)[/tex]-values are repeated with different [tex]\( y \)[/tex]-values.
Therefore, the table represents a function because every [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value.
Given the options:
A. No, because none of the [tex]\( y \)[/tex]-values are the same.
B. No, because each [tex]\( x \)[/tex]-value is different.
C. Yes, because every [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value.
D. Yes, because the [tex]\( y \)[/tex]-values are positive numbers.
The correct answer is:
C. Yes, because every [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value.
