Answer :
Let's carefully analyze the given question and tables in order to draw a proper conclusion.
We have the following tables for \( f(x) \) and \( f^{-1}(x) \):
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline x & 0 & 1 & 2 \\ \hline f(x) & 1 & 10 & 100 \\ \hline \end{tabular} \][/tex]
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline x & 1000 & 100 & 10 \\ \hline f^{-1}(x) & 3 & 2 & 1 \\ \hline \end{tabular} \][/tex]
Let's examine the data step-by-step:
1. Values of \( f(x) \):
- When \( x = 0 \), \( f(x) = 1 \)
- When \( x = 1 \), \( f(x) = 10 \)
- When \( x = 2 \), \( f(x) = 100 \)
So, the function \( f(x) \) gives us the points: (0, 1), (1, 10), and (2, 100). Therefore, the range of \( f(x) \) is the set of all \( y \)-values: \( \{1, 10, 100\} \).
2. Values of \( f^{-1}(x) \):
- When \( x = 1000 \), \( f^{-1}(x) = 3 \)
- When \( x = 100 \), \( f^{-1}(x) = 2 \)
- When \( x = 10 \), \( f^{-1}(x) = 1 \)
So, the inverse function \( f^{-1}(x) \) provides us with the points: (1000, 3), (100, 2), and (10, 1). For \( f^{-1}(x) \), the domain consists of these \( x \)-values: \( \{1000, 100, 10\} \).
3. Relationship Between \( f \) and \( f^{-1} \):
- \( f(x) = y \) implies \( f^{-1}(y) = x \)
- To verify the correctness of this, we should check if the \( y \)-values obtained from \( f(x) \) match the \( x \)-values in \( f^{-1}(x) \) and vice versa:
- The range of \( f(x) \) is \( \{1, 10, 100\} \), and the domain values for \( f^{-1}(x) \) are \( \{1000, 100, 10\} \).
4. Verifying the Domain of \( f^{-1}(x) \):
- To confirm consistency, check that every entry in the range of \( f(x) \) is covered in the domain of \( f^{-1}(x) \). Here, the values \( 1, 10, \) and \( 100 \) are rightly covered by the \( x \)-values found in the table of \( f^{-1}(x) \).
Based on this analysis, we deduce the following conclusion:
The range of \( f(x) \) includes values such that \( y \geq 1 \). These values encompass \( y = 1, 10, \) and \( 100 \). Therefore, the domain of \( f^{-1}(x) \) must include values such that \( x \geq 1 \).
Hence, the accurate conclusion from the given data is:
"The range of [tex]\( f(x) \)[/tex] includes values such that [tex]\( y \geq 1 \)[/tex], so the domain of [tex]\( f^{-1}(x) \)[/tex] includes values such that [tex]\( x \geq 1 \)[/tex]."
We have the following tables for \( f(x) \) and \( f^{-1}(x) \):
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline x & 0 & 1 & 2 \\ \hline f(x) & 1 & 10 & 100 \\ \hline \end{tabular} \][/tex]
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline x & 1000 & 100 & 10 \\ \hline f^{-1}(x) & 3 & 2 & 1 \\ \hline \end{tabular} \][/tex]
Let's examine the data step-by-step:
1. Values of \( f(x) \):
- When \( x = 0 \), \( f(x) = 1 \)
- When \( x = 1 \), \( f(x) = 10 \)
- When \( x = 2 \), \( f(x) = 100 \)
So, the function \( f(x) \) gives us the points: (0, 1), (1, 10), and (2, 100). Therefore, the range of \( f(x) \) is the set of all \( y \)-values: \( \{1, 10, 100\} \).
2. Values of \( f^{-1}(x) \):
- When \( x = 1000 \), \( f^{-1}(x) = 3 \)
- When \( x = 100 \), \( f^{-1}(x) = 2 \)
- When \( x = 10 \), \( f^{-1}(x) = 1 \)
So, the inverse function \( f^{-1}(x) \) provides us with the points: (1000, 3), (100, 2), and (10, 1). For \( f^{-1}(x) \), the domain consists of these \( x \)-values: \( \{1000, 100, 10\} \).
3. Relationship Between \( f \) and \( f^{-1} \):
- \( f(x) = y \) implies \( f^{-1}(y) = x \)
- To verify the correctness of this, we should check if the \( y \)-values obtained from \( f(x) \) match the \( x \)-values in \( f^{-1}(x) \) and vice versa:
- The range of \( f(x) \) is \( \{1, 10, 100\} \), and the domain values for \( f^{-1}(x) \) are \( \{1000, 100, 10\} \).
4. Verifying the Domain of \( f^{-1}(x) \):
- To confirm consistency, check that every entry in the range of \( f(x) \) is covered in the domain of \( f^{-1}(x) \). Here, the values \( 1, 10, \) and \( 100 \) are rightly covered by the \( x \)-values found in the table of \( f^{-1}(x) \).
Based on this analysis, we deduce the following conclusion:
The range of \( f(x) \) includes values such that \( y \geq 1 \). These values encompass \( y = 1, 10, \) and \( 100 \). Therefore, the domain of \( f^{-1}(x) \) must include values such that \( x \geq 1 \).
Hence, the accurate conclusion from the given data is:
"The range of [tex]\( f(x) \)[/tex] includes values such that [tex]\( y \geq 1 \)[/tex], so the domain of [tex]\( f^{-1}(x) \)[/tex] includes values such that [tex]\( x \geq 1 \)[/tex]."
