1. The problem states that the relation $f$ is defined by $f:x-x^2-2$ where $x$ belongs to the set $\{-1,-2,0,2\}$. We need to find $f(x)$ for each $x$ in this set. 2. The formula for $f(x)$ is given by: $$f(x) = x - x^2 - 2.$$ We will substitute each $x$ value into this formula. 3. Calculate $f(-1)$: $$f(-1) = (-1) - (-1)^2 - 2 = -1 - 1 - 2 = -4.$$ 4. Calculate $f(-2)$: $$f(-2) = (-2) - (-2)^2 - 2 = -2 - 4 - 2 = -8.$$ 5. Calculate $f(0)$: $$f(0) = 0 - 0^2 - 2 = 0 - 0 - 2 = -2.$$ 6. Calculate $f(2)$: $$f(2) = 2 - 2^2 - 2 = 2 - 4 - 2 = -4.$$ Final answer: For $x = -1$, $f(x) = -4$. For $x = -2$, $f(x) = -8$. For $x = 0$, $f(x) = -2$. For $x = 2$, $f(x) = -4$.