How can I read the domain and range from a function graph?

Treat the graph as a collection of points. Each point pairs an input, its x-coordinate, with an output, its y-coordinate. The domain is all the x-values used by those points; the range is all their y-values.

For the domain, imagine scanning horizontally from left to right and recording every x-value where the graph exists. For the range, scan vertically from bottom to top and record every y-value reached. In short: domain is the horizontal extent; range is the vertical extent. [S1]

Find the graph’s leftmost and rightmost extents, then its lowest and highest extents. Don’t assume the same points give both pairs of endpoints: a curve can be farthest left at one point and lowest at another. Check for gaps or separate pieces before writing one continuous interval.

An open circle means the endpoint value is excluded; a closed, filled circle means it is included. In interval notation, use a parenthesis for an excluded endpoint and a bracket for an included endpoint. [S1]

Apply that rule to the endpoint value you are recording. For the domain, inspect whether the leftmost or rightmost x-value is included. For the range, inspect whether the lowest or highest y-value is included. A circle’s location matters in both coordinates, but check the whole graph: another part may include the same x- or y-value.

Suppose, just for this example, the graph is a straight line segment that starts at an open circle at (-2, 1) and ends at a closed circle at (3, 4), with no other pieces.

1. Scan left to right. The smallest x-value is -2, but its endpoint is open, so exclude it. The largest x-value is 3, and its endpoint is closed, so include it. The domain is (-2, 3].
2. Scan bottom to top. The smallest y-value is 1, and its endpoint is open, so exclude it. The largest y-value is 4, and its endpoint is closed, so include it. The range is (1, 4].

The segment passes through every intermediate x- and y-value, so these are continuous intervals. For a graph with gaps or separate pieces, describe the values actually covered rather than automatically filling in between the extremes.

Use this sequence: identify each graph piece and any gaps; scan horizontally for the x-values covered; mark the left and right limits and check whether each is included; then scan vertically for the y-values covered and check the lowest and highest limits the same way. Combine intervals only when the graph covers the values between them.

For the example above, a compact written answer is: Domain: (-2, 3]. Range: (1, 4]. If a question asks for set notation rather than interval notation, use the requested format. The graph-reading method identifies the values; the question determines how to present them.

If the picture does not make a value or endpoint clear, don’t guess whether it is included. Check any labels or additional information provided with the problem.