Let's go through the problem step by step using the given table.
Given table:
[tex]\[
\begin{array}{|r|r|}
\hline
x & f(x) \\
\hline
0 & 4 \\
\hline
1 & 8 \\
\hline
2 & 5 \\
\hline
3 & 7 \\
\hline
4 & 0 \\
\hline
5 & 6 \\
\hline
6 & 3 \\
\hline
7 & 9 \\
\hline
8 & 1 \\
\hline
9 & 2 \\
\hline
\end{array}
\][/tex]
1. Find [tex]\( f(2) \)[/tex]:
Look for the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 2 \)[/tex]. According to the table:
[tex]\[
f(2) = 5
\][/tex]
2. Find [tex]\( f^{-1}(6) \)[/tex]:
[tex]\( f^{-1}(6) \)[/tex] means we need to find the value of [tex]\( x \)[/tex] for which [tex]\( f(x) = 6 \)[/tex]. From the table:
[tex]\[
f(5) = 6 \implies f^{-1}(6) = 5
\][/tex]
3. Find [tex]\( f\left(f^{-1}(4)\right) \)[/tex]:
- First, find [tex]\( f^{-1}(4) \)[/tex]. This means finding [tex]\( x \)[/tex] such that [tex]\( f(x) = 4 \)[/tex]. From the table:
[tex]\[
f(0) = 4 \implies f^{-1}(4) = 0
\][/tex]
- Then, find [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = 4 \implies f\left(f^{-1}(4)\right) = 4
\][/tex]
4. If [tex]\( f(x) = 9 \)[/tex], find [tex]\( x \)[/tex]:
We need to find [tex]\( x \)[/tex] such that [tex]\( f(x) = 9 \)[/tex]. According to the table:
[tex]\[
f(7) = 9 \implies x = 7
\][/tex]
Therefore, the filled-in values are:
[tex]\[
\begin{array}{|r|r|}
\hline
x & f(x) \\
\hline
0 & 4 \\
\hline
1 & 8 \\
\hline
2 & 5 \\
\hline
3 & 7 \\
\hline
4 & 0 \\
\hline
5 & 6 \\
\hline
6 & 3 \\
\hline
7 & 9 \\
\hline
8 & 1 \\
\hline
9 & 2 \\
\hline
\end{array}
\][/tex]
[tex]\( f(2) = 5 \)[/tex]
[tex]\[
\begin{array}{l}
f^{-1}(6) = 5 \\
f\left(f^{-1}(4)\right) = 4 \\
\end{array}
\][/tex]
If [tex]\( f(x) = 9 \)[/tex], then [tex]\( x = 7 \)[/tex]
