Understanding the Domain of a Function
The domain of a function represents all the possible input values (often referred to as x values) that the function can accept. To find the domain of a function from its graph, follow these steps:
1. Analyze the Graph
Look closely at the graph you’re given. Start by identifying the horizontal extent of the graph. This means checking which x values are represented by the graph. Is the graph defined for all x values, or does it start and stop at certain points?
2. Identify Gaps or Restrictions
Search for any gaps, jumps, or breaks in the graph. These can indicate that certain x values are not part of the domain. For example, if the graph is undefined at x = 2, then 2 is not included in the domain.
3. Check for Vertical Asymptotes
Vertical asymptotes typically represent values where the function is undefined. For example, if there is a vertical line at x = -1 where the function approaches infinity or negative infinity, then -1 is also excluded from the domain.
4. Consider Open and Closed Circles
On some graphs, you may see circles indicating whether a point is included in the domain or not. A closed circle means that the point is included, while an open circle indicates that it is not. Pay close attention to these symbols as they will directly affect the domain.
5. Write the Domain in Interval Notation
Once you have identified all the valid x values, you can express the domain using interval notation. For example, if the function is defined for all x values from -3 to 4, excluding 2, the domain would be written as:
-3 < x < 2 or 2 < x < 4
This can be simplified in interval notation to:
( -3, 2 ) ∪ ( 2, 4 )
Conclusion
Finding the domain of a function from its graph is a straightforward process. By closely examining the graph for continuity, identifying any restrictions, and writing the domain in clear mathematical terms, you can effectively determine all the values that the function can accept. With practice, this skill will enable you to graph and analyze other functions with greater confidence!