To determine the inverse of the function [tex]\( f \)[/tex] denoted by [tex]\( f^{-1} \)[/tex], we can use the provided function values table. The goal is to find the [tex]\( x \)[/tex]-values for specific [tex]\( f(x) \)[/tex] values and vice versa. Here is the step-by-step process:
1. Identify each [tex]\( f(x) \)[/tex] value in the given table and find the corresponding [tex]\( x \)[/tex]-value.
Given the function table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -2 & -1 & 0 & 1 & 2 \\
\hline
f(x) & -28 & -9 & -2 & -1 & 0 \\
\hline
\end{array}
\][/tex]
We can write these mappings:
- [tex]\( f(-2) = -28 \)[/tex]
- [tex]\( f(-1) = -9 \)[/tex]
- [tex]\( f(0) = -2 \)[/tex]
- [tex]\( f(1) = -1 \)[/tex]
- [tex]\( f(2) = 0 \)[/tex]
2. Find [tex]\( f^{-1}(x) \)[/tex] by reversing these mappings, i.e., for a given [tex]\( f(x) \)[/tex], determine the corresponding [tex]\( x \)[/tex]-value:
- [tex]\( f^{-1}(-28) = -2 \)[/tex]
- [tex]\( f^{-1}(-9) = -1 \)[/tex]
- [tex]\( f^{-1}(-2) = 0 \)[/tex]
- [tex]\( f^{-1}(-1) = 1 \)[/tex]
- [tex]\( f^{-1}(0) = 2 \)[/tex]
3. Now we fill in the table for [tex]\( f^{-1} \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -28 & -9 & -2 & -1 & 0 \\
\hline
f^{-1}(x) & -2 & -1 & 0 & 1 & 2 \\
\hline
\end{array}
\][/tex]
Using these values, we can complete the provided partial table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -2 & -1 & 0 & -1 & 0 \\
\hline
f^{-1}(x) & -2 & 1 & 0 & 1 & 2 \\
\hline
\end{array}
\][/tex]
So, the correct compositions should be:
\begin{array}{|c|c|c|c|c|}
\hline
x & -9 & -2 & -1 & 0 \\
\hline
f^{-1}(x) & -1 & 0 & 1 & 2 \\
\hline
\end{array}
