Determine the range of a function from a graph

Detailed Explanation: Determine the range of a function from a graph

To determine the range, look at the graph’s possible $y$-values.

Think of sliding a horizontal line up and down the graph:

• If the line touches the graph, that $y$-value is in the range.
• If it never touches the graph, that $y$-value is not in the range.
• A filled dot means the endpoint value is included.
• An open circle means the endpoint value is excluded.
• An isolated point adds its own $y$-value to the range.

Example

Suppose a graph contains:

• a line segment from a filled point at $((-3,-2))$ to an open point at $((2,3))$, and
• an isolated filled point at $((4,5))$.

We want to find the range.

Step 1: Trace the graph vertically

The line segment reaches every $y$-value from $(-2)$ up to $3$.

So far, the possible outputs are between $(-2)$ and $3$.

Step 2: Check the endpoints

• The point at $(y=-2)$ is filled, so $(-2)$ is included.
• The point at $(y=3)$ is open, so $3$ is not included.

This gives the interval

Step 3: Include isolated points

The isolated filled point is at $(y=5)$, so $5$ is also in the range.

Therefore, the range is

In set-builder notation, this is

Remember: the range lists the $y$-values, not the $x$-values.