1. **Problem Statement:** We are given a function $f(x)$ with inputs and outputs in a table and asked to complete the table for its inverse function $f^{-1}(x)$. 2. **Understanding the inverse function:** The inverse function $f^{-1}(x)$ reverses the roles of inputs and outputs of $f(x)$. That means if $f(a) = b$, then $f^{-1}(b) = a$. 3. **Given table:** $$\begin{array}{c|cccccc} x & -7 & 11 & -13 & 6 & 5 & -9 \\ f(x) & 7 & 12 & 8 & -7 & 13 & 5 \\\end{array}$$ 4. **Complete the inverse table:** Swap the rows and columns for $f^{-1}(x)$: - Since $f(-7) = 7$, then $f^{-1}(7) = -7$ - Since $f(11) = 12$, then $f^{-1}(12) = 11$ - Since $f(-13) = 8$, then $f^{-1}(8) = -13$ - Since $f(6) = -7$, then $f^{-1}(-7) = 6$ - Since $f(5) = 13$, then $f^{-1}(13) = 5$ - Since $f(-9) = 5$, then $f^{-1}(5) = -9$ So the inverse table is: $$\begin{array}{c|cccccc} x & 7 & 12 & 8 & -7 & 13 & 5 \\ f^{-1}(x) & -7 & 11 & -13 & 6 & 5 & -9 \\\end{array}$$ 5. **Find the requested values:** - $f^{-1}(f^{-1}(13))$: - First, $f^{-1}(13) = 5$ - Then, $f^{-1}(5) = -9$ - So, $f^{-1}(f^{-1}(13)) = -9$ - $f^{-1}(8)$: - From the table, $f^{-1}(8) = -13$ - $f^{-1}(f(10))$: - We need $f(10)$, but $10$ is not in the domain table, so we cannot find $f(10)$ from the given data. - Since $f(10)$ is unknown, $f^{-1}(f(10))$ cannot be determined from the given information. 6. **Matching function with inverse type:** - The function $f(x) = 3x - 5$ is a linear function, not quadratic. - Its inverse is also linear, given by solving $y = 3x - 5$ for $x$: $$x = \frac{y + 5}{3}$$ **Final answers:** - Completed inverse table: $$\begin{array}{c|cccccc} x & 7 & 12 & 8 & -7 & 13 & 5 \\ f^{-1}(x) & -7 & 11 & -13 & 6 & 5 & -9 \\\end{array}$$ - $f^{-1}(f^{-1}(13)) = -9$ - $f^{-1}(8) = -13$ - $f^{-1}(f(10))$ cannot be determined from the given data. - $f(x) = 3x - 5$ is a linear function, so its inverse is linear, not quadratic.