1. **Problem Statement:** Determine if the given relations are functions by analyzing their domain, range, and the vertical line test. 2. **Key Concept:** A relation is a function if every element in the domain (independent variable $x$) corresponds to exactly one element in the range (dependent variable $y$). 3. **Vertical Line Test:** If a vertical line intersects the graph at more than one point, the relation is not a function. 4. **Example 1:** Given points $(-3,2), (-2,-1), (2,1), (2,-3)$ - Domain: $x = \{-3, -2, 2\}$ - Range: $y = \{-3, -1, 1, 2\}$ - Since $x=2$ corresponds to two different $y$ values ($1$ and $-3$), this fails the vertical line test. **Conclusion:** This relation is **not a function**. 5. **Example 2:** Domain: $x = \{-3, -1, 1, 2\}$, Range: $y = \{-3, 0, 2\}$ - Each $x$ has exactly one $y$. **Conclusion:** This relation **is a function**. 6. **Example 3:** Domain: $x \geq 3$, Range: $y \leq 2$ - The graph is a line with negative slope passing through $(-3,2)$ and extending rightwards. - For each $x$ in the domain, there is exactly one $y$. **Conclusion:** This relation **is a function**. 7. **Example 4:** Domain: all real numbers $\mathbb{R}$, Range: all real numbers $\mathbb{R}$ - The graph represents a function defined for all real $x$. **Conclusion:** This relation **is a function**. 8. **Example 5:** Domain: $-2 \leq x \leq 1$, Range: $-2 \leq y \leq 3$ - The graph is a line segment between points $(-2,3)$ and $(1,-2)$. - Each $x$ in the domain corresponds to exactly one $y$. **Conclusion:** This relation **is a function**. **Summary:** - Relations where a single $x$ maps to multiple $y$ values are not functions. - Relations passing the vertical line test are functions.