To find the inverse of the function [tex]\( f \)[/tex], we swap the roles of [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex]. This means we use the values of [tex]\( f(x) \)[/tex] as the new [tex]\( x \)[/tex] values and the original [tex]\( x \)[/tex] values as [tex]\( f^{-1}(x) \)[/tex].
Given:
[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 need to complete the table for the inverse function [tex]\( f^{-1} \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & \square & \square & \square & -1 & 0 \\
\hline
f^{-1}(x) & -2 & \square & 0 & \square & \square \\
\hline
\end{array}
\][/tex]
By swapping the given table values, we get:
[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]
Now match these values with the required boxes:
- For the [tex]\( x \)[/tex]-row:
- [tex]\( f^{-1}(-28) = -2 \)[/tex]
- [tex]\( f^{-1}(-9) = -1 \)[/tex]
- [tex]\( f^{-1}(-2) = 0 \)[/tex]
- For the [tex]\( f^{-1} (x) \)[/tex]-row:
- [tex]\( f^{-1}(-1) = 1 \)[/tex]
- [tex]\( f^{-1}(0) = 2 \)[/tex]
Thus, the completed inverse function table is:
[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]
