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Sagot :
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.
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.
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