1. **State the problem:** We need to graph the piecewise function defined as: $$f(x) = \begin{cases} 2 & \text{if } 2 < x \leq 4 \\ x + 3 & \text{if } 4 < x < 8 \\ 2x & \text{if } x \geq 8 \end{cases}$$ 2. **Analyze each piece:** - For $2 < x \leq 4$, $f(x) = 2$ is a constant function, so the graph is a horizontal line at $y=2$ between $x=2$ (not included) and $x=4$ (included). - For $4 < x < 8$, $f(x) = x + 3$ is a linear function with slope 1 and y-intercept 3, defined on the open interval $(4,8)$. - For $x \geq 8$, $f(x) = 2x$ is a linear function with slope 2 and y-intercept 0, defined for $x$ starting at 8 and beyond. 3. **Evaluate key points:** - At $x=4$, from the first piece, $f(4) = 2$ (included). - At $x=4$, from the second piece, $f(4) = 4 + 3 = 7$ (not included since $4$ is not in $(4,8)$). - At $x=8$, from the second piece, $f(8) = 8 + 3 = 11$ (not included since $8$ is not in $(4,8)$). - At $x=8$, from the third piece, $f(8) = 2 \times 8 = 16$ (included). 4. **Summary of graph behavior:** - Horizontal line at $y=2$ from just after $x=2$ to $x=4$ inclusive. - Open circle at $(4,7)$ and $(8,11)$ for the second piece. - Line starting at $(8,16)$ and increasing with slope 2 for $x \geq 8$. 5. **Final answer:** The piecewise function is graphed as described above with the specified intervals and points.