To solve the problem of finding [tex]\( f^{-1}(10) \)[/tex] using the given table, we'll follow these detailed steps:
1. Understand the Function:
We are given a table representing a linear function [tex]\( f(x) \)[/tex]. We need to find the value of [tex]\( x \)[/tex] for which [tex]\( f(x) = 10 \)[/tex]. This inverse function is denoted as [tex]\( f^{-1}(10) \)[/tex].
2. Inspect the Table:
The table provides values of [tex]\( x \)[/tex] and the corresponding [tex]\( f(x) \)[/tex]. Let's write down the given pairs for clarity:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -4 & 0 & 2 & 5 & 9 & 10 \\
\hline
f(x) & -11 & 1 & 7 & 16 & 28 & 31 \\
\hline
\end{array}
\][/tex]
3. Analyze the Target Value:
We need to find which [tex]\( x \)[/tex] value gives [tex]\( f(x) = 10 \)[/tex].
4. Search for [tex]\( f(x) = 10 \)[/tex]:
Look through the [tex]\( f(x) \)[/tex] values in the table:
- For [tex]\( x = -4 \)[/tex], [tex]\( f(x) = -11 \)[/tex]
- For [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 1 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 7 \)[/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\( f(x) = 16 \)[/tex]
- For [tex]\( x = 9 \)[/tex], [tex]\( f(x) = 28 \)[/tex]
- For [tex]\( x = 10 \)[/tex], [tex]\( f(x) = 31 \)[/tex]
Observe that none of these [tex]\( f(x) \)[/tex] values is equal to 10.
5. Conclusion:
Since none of the given [tex]\( f(x) \)[/tex] values equals 10, it indicates that there is no [tex]\( x \)[/tex] in the provided table for which [tex]\( f(x) = 10 \)[/tex].
Therefore, the solution is:
[tex]\[ f^{-1}(10) = \text{None} \][/tex]
None of the values provided as options (31, 28, 3, 9) are correct, since [tex]\( f(x) = 10 \)[/tex] does not exist in the table.
