1. **Problem statement:** We have a piecewise function defined as: $$h(x) = \begin{cases} -\frac{x}{2} & \text{if } x \neq 3 \\ 5 & \text{if } x = 3 \end{cases}$$ We want to understand how to graph and interpret this function. 2. **What is a piecewise function?** It means the function behaves differently depending on the value of $x$. Here, for every $x$ except 3, $h(x)$ follows the line $y = -\frac{x}{2}$. But at $x=3$, the function jumps to $y=5$. 3. **Graphing the line part:** The line $y = -\frac{x}{2}$ means for each $x$, you take half of $x$, then make it negative. For example: - If $x=0$, $y = -\frac{0}{2} = 0$ - If $x=2$, $y = -\frac{2}{2} = -1$ - If $x=4$, $y = -\frac{4}{2} = -2$ 4. **What happens at $x=3$?** Normally, the line would give $y = -\frac{3}{2} = -1.5$. But the function says at $x=3$, $h(3) = 5$ instead. 5. **Open and closed circles on the graph:** - At $x=3$, the point on the line $y = -1.5$ is shown as an open circle. This means the function does NOT include this value. - Instead, there is a closed (filled) circle at $(3,5)$ showing the actual function value there. 6. **Summary:** - The graph is mostly the line $y = -\frac{x}{2}$. - At $x=3$, the function jumps to $y=5$. - This is why the graph has an open circle on the line at $x=3$ and a filled circle at $(3,5)$. This helps us understand that piecewise functions can have "holes" or jumps at certain points. **Final answer:** The function $h(x)$ is the line $y = -\frac{x}{2}$ everywhere except at $x=3$, where it equals 5, shown by a filled dot at $(3,5)$ and an open circle on the line at $(3,-1.5)$.