1. **Problem Statement:** Graph the greatest integer function $f(x) = \lfloor x \rfloor$, which outputs the greatest integer less than or equal to $x$. 2. **Understanding the Function:** The function $f(x) = \lfloor x \rfloor$ is a step function that remains constant between consecutive integers. 3. **Key Property:** For any $x$ in the interval $[n, n+1)$ where $n$ is an integer, $f(x) = n$. 4. **Graphing Steps:** - Between $x = -2$ and $x = -1$, $f(x) = -2$. This is a horizontal line segment at $y = -2$. - At $x = -2$, the point is included (solid circle) because $f(-2) = -2$. - At $x = -1$, the point is not included (open circle) because $f(-1)$ jumps to $-1$. - This pattern repeats for all integer intervals. 5. **Summary:** The graph consists of horizontal segments from each integer $n$ to $n+1$ at height $y = n$, with solid circles at the left endpoints and open circles at the right endpoints. **Final answer:** The graph of $f(x) = \lfloor x \rfloor$ is a step function with horizontal segments on intervals $[n, n+1)$ at height $n$, solid circles at $x=n$, and open circles at $x=n+1$ for all integers $n$.