Determining the domain of a function from its graph can be easy if you know what to look for. The domain is all about the possible input values, which are usually the x-values, for which the function works. Here’s how to analyze the graph to find the domain:

1. Look at the x-axis: Start by checking how far the graph goes from left to right. The domain will include all the x-values where the graph exists.

2. Spot gaps or breaks: Sometimes, the graph might have gaps where the function isn’t defined. For example, if there’s a hole or a vertical line where the graph goes off to infinity, you should leave those x-values out of the domain.

3. Check endpoints and intervals: If the graph is a line or a curve that keeps going, look for endpoints. If the graph stops at a certain point (like with a closed circle), that endpoint is included in the domain. But if there’s an open circle (a point that the graph doesn’t actually touch), that endpoint isn’t included.

4. Writing the domain: After you find out which x-values are included and which ones aren’t, you can write the domain clearly. You can use interval notation for this! For example:

   • If the graph includes all x-values from -3 to 2, and both ends are included, you would write the domain as $[-3, 2]$.
   • If there’s a gap at x = 1 (meaning the graph jumps from 0.9 to 1.1), you’d write it as $[-3, 1) \cup (1, 2]$.

5. Real-life examples: Sometimes, understanding the domain can involve real-life situations. For example, if a graph shows a person's height over time, negative time values wouldn’t make sense, so you’d leave them out.

To sum it up, analyzing a function’s graph for its domain means checking how far the graph stretches, looking for holes or limits, confirming endpoints, and writing your findings in a clear way. It can feel a bit like a scavenger hunt, as you search for clues to see where the function makes sense!