1. The problem asks to analyze the graph of a function and answer specific questions about values of $x$ and $y$. 2. (a) For what values of $x$ is $y=1$? - From the graph description, $y=1$ occurs near $x=-3$ (since $y$ starts near 1.5 at $x=-3$ and the curve crosses $y=1$ going down) and again near $x=2$ (since the curve rises steeply to about $y=3$ at $x=2.5$ and must cross $y=1$ on the way up). - Approximate values: $x \approx -2.8$ and $x \approx 2$. 3. (b) For what values of $x$ is $y=3$? - The graph reaches $y=3$ at about $x=-2.5$ and again at $x=2.5$. - So $x = -2.5$ and $x = 2.5$. 4. (c) For what values of $y$ is $x=3$? - The graph is only described up to $x=3$, but no specific $y$ value is given at $x=3$. - Since the curve ends near $x=3$ and $y$ is about 1.5 at $x=-3$, by symmetry and smoothness, approximate $y \approx 1.5$ at $x=3$. 5. (d) For what values of $x$ is $y \leq 0$? - The curve dips below the $x$-axis around $x=-1.5$ and again near $x=1$. - So $y \leq 0$ approximately for $x$ in the intervals $[-1.5,0]$ and $[0.5,1.5]$ (estimating from the description). 6. (e) What are the maximum and minimum values of $y$ and for what values of $x$ do they occur? - Maximum $y$ is about 3 at $x=-2.5$ and $x=2.5$. - Minimum $y$ is slightly below $-2$ near $x=1$. Summary: - $y=1$ at $x \approx -2.8, 2$ - $y=3$ at $x = -2.5, 2.5$ - $x=3$ at $y \approx 1.5$ - $y \leq 0$ for $x$ near $[-1.5,0]$ and $[0.5,1.5]$ - Max $y=3$ at $x=-2.5, 2.5$ - Min $y \approx -2$ near $x=1$. 7. Next, determine if each graph defines $y$ as a function of $x$: - (a) Increasing concave function passing origin: passes vertical line test, so yes. - (b) Symmetrical bell-shaped curve: also passes vertical line test, so yes. - (c) Descending curve through origin: passes vertical line test, so yes. - (d) Closed oval (ellipse): fails vertical line test (vertical line intersects twice), so no. Final answers: (a) Yes (b) Yes (c) Yes (d) No