How to Find Domain and Range from a Graph

When you study algebra and functions, it's important to know about domain and range.

• The domain is all the possible input values (that's usually the $x$ values) that a function can take.

• The range is all the possible output values (which are usually the $y$ values) that the function can give.

Let’s go through how to find both the domain and range from a graph.

Finding the Domain

1. Look at the Graph Horizontally:

   • Check the graph from side to side.
   • The domain includes the $x$ values where the graph exists.
   • Find out where the graph starts and ends along the $x$ axis. If the graph goes on forever to the left or right, the domain can also be infinite.

2. Check for Restrictions:

   • Look for things like holes, vertical lines that the graph gets close to (called asymptotes), or places where the function can’t work (like when a denominator is zero).
   • For example, if there’s a vertical line at $x = 2$, then $2$ is not part of the domain.

3. Write the Domain:

   • Use interval notation. For example, if the domain goes from $-3$ to $4$, but skips $2$, you would write it like this: $(-3, 2) \cup (2, 4)$.
   • If the graph covers all real numbers, you can write the domain as $(-\infty, \infty)$.

Finding the Range

1. Look at the Graph Vertically:

   • Check the graph from top to bottom.
   • The range is about the $y$ values that the function produces as you read the graph from left to right.
   • Find the lowest and highest points on the graph. If it goes up or down forever, the range could also be infinite.

2. Identify Important Points:

   • Look for peaks (high points) and valleys (low points) on the graph. These points help define the range.
   • For example, if the lowest point is $y = -1$ and the graph goes up forever, you write the range as $[-1, \infty)$.

3. Write the Range:

   • Use interval notation again. If the range goes from $-1$ to forever, you can write it as $[-1, \infty)$.
   • If it ranges from $0$ to $3$, you write it as $[0, 3]$.

Steps to Remember

To Find the Domain:

• Look horizontally to see where the graph is along the $x$ axis.
• Identify and remove any $x$ values that are not allowed.
• Use interval notation to show the domain.

To Find the Range:

• Look vertically to see where the graph is along the $y$ axis.
• Find the maximum and minimum $y$ values.
• Use interval notation to show the range.

Conclusion

Knowing how to find the domain and range from a graph is a key skill in Algebra I. By looking at the graph from side to side and top to bottom, and by spotting important points or restrictions, you can figure out the input and output values of functions. Mastering these ideas helps you move on to more difficult math topics.