To find the inverse of the given function [tex]\( f(x) \)[/tex], we first need to understand how inverses work. The inverse function [tex]\( f^{-1}(x) \)[/tex] essentially swaps the roles of the inputs (x) and the outputs [tex]\( f(x) \)[/tex]. For each pair [tex]\( (x, f(x)) \)[/tex] in the given function, the inverse function will have the pair [tex]\( (f(x), x) \)[/tex].
Let's determine the inverse function step-by-step using the given function values:
### Original Function
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & 9 \\
\hline
-1 & 7 \\
\hline
0 & 5 \\
\hline
1 & 3 \\
\hline
2 & 1 \\
\hline
\end{array}
\][/tex]
### Inverse Function Construction
1. For [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 9 \)[/tex]. Thus, in the inverse function, we have [tex]\( f^{-1}(9) = -2 \)[/tex].
2. For [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 7 \)[/tex]. Thus, in the inverse function, we have [tex]\( f^{-1}(7) = -1 \)[/tex].
3. For [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 5 \)[/tex]. Thus, in the inverse function, we have [tex]\( f^{-1}(5) = 0 \)[/tex].
4. For [tex]\( x = 1 \)[/tex], [tex]\( f(x) = 3 \)[/tex]. Thus, in the inverse function, we have [tex]\( f^{-1}(3) = 1 \)[/tex].
5. For [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 1 \)[/tex]. Thus, in the inverse function, we have [tex]\( f^{-1}(1) = 2 \)[/tex].
Putting all these together, the inverse function [tex]\( f^{-1}(x) \)[/tex] can be represented as follows:
[tex]\[
\begin{array}{|c|c|}
\hline
f(x) & x \\
\hline
9 & -2 \\
\hline
7 & -1 \\
\hline
5 & 0 \\
\hline
3 & 1 \\
\hline
1 & 2 \\
\hline
\end{array}
\][/tex]
### Inverse Function Table Representation
[tex]\[
\begin{array}{|c|c|}
\hline
y & f^{-1}(y) \\
\hline
9 & -2 \\
\hline
7 & -1 \\
\hline
5 & 0 \\
\hline
3 & 1 \\
\hline
1 & 2 \\
\hline
\end{array}
\][/tex]
Therefore, the inverse of the given function is:
[tex]\[
\{ 9: -2, 7: -1, 5: 0, 3: 1, 1: 2 \}
\][/tex]
This provides us with [tex]\( f^{-1}(y) \)[/tex] for each value of [tex]\( y \)[/tex] corresponding to the function values of the original function.
