1. **Problem:** Determine if the set defined by $\{(x,y): x = y^2\}$ is a function or just a relation. 2. **Recall:** A relation is a function if for every $x$ there is exactly one $y$. 3. **Analyze:** Here, $x = y^2$ means for each $x$, $y = \pm \sqrt{x}$ (two values of $y$ for $x > 0$). 4. **Conclusion:** Since one $x$ corresponds to two $y$ values, this is a relation but not a function. 2. **Problem:** Determine if $\{(x,y): x^2 + y^2 = 1\}$ is a function. 3. **Analyze:** This is a circle. For many $x$ values, there are two $y$ values (top and bottom of the circle). 4. **Conclusion:** Not a function, only a relation. 3. **Problem:** Determine if $\{(x,y): y = \sqrt{4 - x^2}\}$ is a function. 4. **Analyze:** For each $x$ in $[-2,2]$, $y$ is the positive root only. 5. **Conclusion:** Each $x$ has exactly one $y$, so this is a function. 4. **Problem:** Determine if $\{(x,y): 2y = 4x^2 - 2x + 6\}$ is a function. 5. **Rewrite:** $y = \frac{4x^2 - 2x + 6}{2} = 2x^2 - x + 3$. 6. **Analyze:** For each $x$, there is exactly one $y$. 7. **Conclusion:** This is a function. 5. **Problem:** Determine if $\{(x,y): y = |2x - 6|\}$ is a function. 6. **Analyze:** Absolute value outputs one value for each $x$. 7. **Conclusion:** This is a function. 6. **Problem:** Determine if $\{(x,y): xy + 5x = 3y - 9\}$ is a function. 7. **Rewrite:** $xy - 3y = -5x - 9$. 8. **Factor:** $y(x - 3) = -5x - 9$. 9. **Solve for $y$:** $y = \frac{-5x - 9}{x - 3}$, $x \neq 3$. 10. **Analyze:** For each $x \neq 3$, exactly one $y$. 11. **Conclusion:** This is a function on domain $x \neq 3$. **Final answers:** 1. Relation only 2. Relation only 3. Function 4. Function 5. Function 6. Function (except at $x=3$ where undefined)