1. Problem statement: Determine whether each given graph (a) through (d) represents $y$ as a function of $x$ by applying the vertical line test. 2. Formula and rule: A relation is a function when each input $x$ has exactly one output $y$. $$\forall x\ \exists!\ y:\ y=f(x)$$ Important rule: The vertical line test says the graph represents a function iff no vertical line intersects the graph more than once. 3. Graph (a) analysis: Description: the curve starts near $(0,0)$ and then splits into two branches, so some vertical lines intersect the graph in two points. Example: a vertical line near $x=0$ meets both branches, giving two different $y$ values for the same $x$. Conclusion: Graph (a) is not a function. 4. Graph (b) analysis: Description: two line segments, one from $(-2,-4)$ to $(0,0)$ and the other from $(1,2)$ to $(3,6)$, with a gap between them. Compute slopes: first segment slope $m=\frac{0-(-4)}{0-(-2)}=\frac{4}{2}=2$. Second segment slope $m=\frac{6-2}{3-1}=\frac{4}{2}=2$. Because for any $x$ in the domain $[-2,0]\cup[1,3]$ a vertical line meets the graph at most once, Graph (b) is a function (piecewise linear). 5. Graph (c) analysis: Description: a V shape composed of a segment from $(0,2)$ down to $(1,0)$ and a segment from $(1,0)$ up to $(3,2)$. Slopes: left slope $m=\frac{0-2}{1-0}=\frac{-2}{1}=-2$. Right slope $m=\frac{2-0}{3-1}=\frac{2}{2}=1$. Every vertical line intersects the V at most once, so Graph (c) is a function. 6. Graph (d) analysis: Description: a vertical line at $x=1$ from $y=-2$ to $y=2$. A vertical line at $x=1$ meets the graph at many points (more than one), so it fails the vertical line test and Graph (d) is not a function. Final answers: a: Not a function. b: Function. c: Function. d: Not a function.