1. **Problem Statement:** Determine the domain and range of the relation defined by the vector $\vec{AB} = -4\hat{i} + 2\hat{j}$ where point A is at (-1, -1) and point B is at (3, 1). Also, state whether this relation is a function. 2. **Understanding the Vector:** The vector $\vec{AB}$ represents the displacement from point A to point B. It has components: $$\vec{AB} = (3 - (-1))\hat{i} + (1 - (-1))\hat{j} = 4\hat{i} + 2\hat{j}$$ Note: The user’s vector $-4\hat{i} + 2\hat{j}$ seems to be the vector from B to A, which is the negative of $\vec{AB}$. 3. **Domain and Range:** - The domain is the set of all possible $x$-values. - The range is the set of all possible $y$-values. Since the vector represents a single displacement from A to B, the relation consists of two points: - Point A: $(-1, -1)$ - Point B: $(3, 1)$ Thus: - Domain: $\{-1, 3\}$ - Range: $\{-1, 1\}$ 4. **Is it a function?** A function assigns exactly one $y$ value to each $x$ value. Here, the domain values $-1$ and $3$ each correspond to exactly one $y$ value: - $x = -1 \to y = -1$ - $x = 3 \to y = 1$ Therefore, this relation is a function. **Final answer:** - Domain: $\{-1, 3\}$ - Range: $\{-1, 1\}$ - The relation is a function.