1. **State the problem:** We need to evaluate compositions of functions $f$ and $g$ using the given table values for $x = \{-2, -1, 0, 1, 2\}$. The functions are defined as: $$\begin{array}{c|ccccc} x & -2 & -1 & 0 & 1 & 2 \\ f(x) & 1 & 0 & -2 & 1 & 2 \\ g(x) & 2 & 1 & 0 & -1 & 0 \\\end{array}$$ We will find values for: (a) $f(g(-1))$ (b) $g(f(0))$ (c) $f(f(-1))$ (d) $g(g(2))$ (e) $g(f(-2))$ (f) $f(g(1))$ 2. **Formula and rules:** To evaluate $f(g(a))$, first find $g(a)$ from the table, then use that result as input to $f$. Similarly for $g(f(a))$, find $f(a)$ first, then use that as input to $g$. This is function composition. 3. **Step-by-step evaluation:** (a) $f(g(-1))$: - Find $g(-1)$ from the table: $g(-1) = 1$ - Now find $f(1)$: $f(1) = 1$ - So, $f(g(-1)) = 1$ (b) $g(f(0))$: - Find $f(0)$: $f(0) = -2$ - Find $g(-2)$: $g(-2) = 2$ - So, $g(f(0)) = 2$ (c) $f(f(-1))$: - Find $f(-1)$: $f(-1) = 0$ - Find $f(0)$: $f(0) = -2$ - So, $f(f(-1)) = -2$ (d) $g(g(2))$: - Find $g(2)$: $g(2) = 0$ - Find $g(0)$: $g(0) = 0$ - So, $g(g(2)) = 0$ (e) $g(f(-2))$: - Find $f(-2)$: $f(-2) = 1$ - Find $g(1)$: $g(1) = -1$ - So, $g(f(-2)) = -1$ (f) $f(g(1))$: - Find $g(1)$: $g(1) = -1$ - Find $f(-1)$: $f(-1) = 0$ - So, $f(g(1)) = 0$ 4. **Final answers:** (a) 1 (b) 2 (c) -2 (d) 0 (e) -1 (f) 0 This method uses direct lookup and substitution to evaluate compositions of discrete functions.