Reading a Domain from a Graph
A graph displays the input-output pairs of a function as points in the coordinate plane. To read the domain, focus on the horizontal coordinates: every input represented by at least one plotted point belongs to the domain. In other words, imagine projecting the graph straight down onto the horizontal axis. The set of horizontal positions reached by that projection is the domain.
This method is useful when a graph is given without a formula or when a formula is not the quickest way to identify the inputs. The details matter: arrows show that a graph continues, gaps leave out inputs, and open and closed endpoints determine whether boundary values are included.
The Horizontal Projection Method
Picture a vertical line through each point of the graph extending down to the horizontal axis. A value \(x=a\) is in the domain exactly when the graph contains a point whose horizontal coordinate is \(a\). The height of that point does not affect whether \(a\) is an input; only its horizontal position matters.
When a graph has several pieces, project every piece onto the horizontal axis and combine the resulting sets. If the pieces project onto overlapping intervals, include their shared inputs only once. If there is a gap between their projections, the inputs in that gap are not in the domain.
What Endpoints and Arrows Tell You
A closed dot marks a plotted point, so its horizontal coordinate is included in the domain. An open circle marks a point that is not part of the graph, so that endpoint’s horizontal coordinate is excluded, unless another plotted point on the graph has that same horizontal coordinate.
On a graph of a function, two different plotted points cannot have the same horizontal coordinate: that would assign two outputs to one input and fail the vertical line test. Thus, for a function graph, an open endpoint with no other plotted point at that horizontal position means that input is excluded.
An arrow indicates that the graph continues in the direction it points. If a branch has an arrow pointing to the right, its horizontal coordinates continue without a right-hand endpoint; if it points to the left, they continue without a left-hand endpoint. A graph that simply reaches the edge of the displayed window does not necessarily stop there. Look for an arrow or other clear indication of an endpoint before concluding that the graph ends.
| Graph feature | What it tells you about the domain |
|---|---|
| Closed endpoint at \(x=a\) | The input \(a\) is included. |
| Open endpoint at \(x=a\) | The input \(a\) is excluded unless another point on the graph has that coordinate. |
| Arrow pointing right or left | The graph continues toward larger or smaller horizontal coordinates, respectively. |
| Gap with no plotted point above an input | That input is not in the domain. |
| Isolated plotted point at \(x=a\) | The input \(a\) is included, even if nearby inputs are not. |
From the Graph to Notation
After identifying the horizontal positions, express them in a useful form. Interval notation describes continuous stretches of inputs. A square bracket includes an endpoint, while a parenthesis excludes it. The symbols \(-\infty\) and \(\infty\) are never endpoints that belong to an interval, so they always appear with parentheses. Separate pieces of a domain are joined using the union symbol \(\cup\).
A domain can also contain isolated inputs. For example, a graph may include one point at \(x=2\) even though it has no plotted points for inputs near \(2\). In that case, use the singleton set \(\{2\}\) and combine it with any intervals that arise from other pieces.
Worked Example: One Segment with Different Endpoint Markers
Suppose a graph consists of a single curve segment that starts at an open circle with coordinates \((-3,2)\) and ends at a closed dot with coordinates \((4,-1)\). The curve includes all points between these endpoints.
First, take the horizontal coordinate of each endpoint. The leftmost input is \(-3\), and the rightmost input is \(4\). The open circle at \(x=-3\) excludes that input. The closed dot at \(x=4\) includes it. Every horizontal coordinate strictly between \(-3\) and \(4\) occurs along the segment.
Check: The interval \((-3,4]\) excludes \(-3\), includes \(4\), and includes every input between them. These are exactly the horizontal positions reached by the segment.
Complete Justification of the Projection Method
The horizontal projection method follows directly from what a function graph represents. Let \(G\) be the graph of \(f\). By definition, its points have the form \((x,f(x))\), where \(x\) is an allowed input. The horizontal coordinate of each such point is therefore an input in the domain. Conversely, if \(x\) belongs to the domain, then \(f(x)\) is defined, so the graph contains the point \((x,f(x))\). Its horizontal coordinate is \(x\). These two implications show that the horizontal coordinates appearing in \(G\) are exactly the inputs in the domain.
Projecting onto the horizontal axis records precisely those horizontal coordinates and discards the vertical coordinates. That is why the method works—and why inspecting only how high or low the graph sits cannot determine its domain.
Combining Pieces, Arrows, and Isolated Points
When a graph contains more than one branch or isolated point, find the horizontal projection of each piece separately. Translate each projection into interval or set notation, then take their union. An open endpoint remains excluded even when the graph approaches it from nearby inputs; approaching a value does not put that value into the domain. A closed point at a separate location, however, contributes its horizontal coordinate even if there are gaps on either side.
Worked Example: Two Branches and an Isolated Point
Suppose a function graph has three pieces: a left branch with an arrow extending to the left and an open endpoint at \((1,2)\); a single closed point at \((1,-2)\); and a right branch beginning at a closed endpoint \((3,0)\) and continuing to the right with an arrow.
The left branch covers every input less than \(1\), so its projection is \((-\infty,1)\). The isolated closed point has horizontal coordinate \(1\), so it adds the set \(\{1\}\). The right branch begins at the included input \(3\) and continues to larger inputs, giving \([3,\infty)\).
Check: All inputs less than \(1\) occur on the left branch, and \(1\) occurs at the isolated closed point. No graph point has a horizontal coordinate strictly between \(1\) and \(3\). The right branch includes \(3\) and continues for every larger input. The union therefore includes exactly the inputs described by the simplified answer.
Worked Example: Separate Intervals and a Missing Endpoint
Suppose a graph consists of a segment whose horizontal coordinates run from \(-4\) at a closed endpoint to \(-2\) at an open endpoint, two isolated closed points at \(x=-1\) and \(x=0\), and a rightward branch with an arrow beginning at an open endpoint at \(x=2\).
The segment projects onto \([-4,-2)\): it includes \(-4\), excludes \(-2\), and includes the inputs between them. The two isolated points contribute \(\{-1\}\) and \(\{0\}\). The rightward branch projects onto \((2,\infty)\), because the endpoint at \(2\) is open and the arrow shows continuation toward larger inputs.
Check: The graph has no points for inputs between \(-2\) and \(-1\), between \(-1\) and \(0\), or between \(0\) and \(2\). The endpoint \(-2\) is excluded, while \(-1\) and \(0\) are included as isolated inputs. The right branch does not include \(2\) but includes every input greater than \(2\). The displayed union matches these features.
A Common Endpoint Mistake
A frequent error is to include an open endpoint just because the curve gets arbitrarily close to it. For example, if a graph has an open circle at \(x=-2\) and a closed endpoint at \(x=3\), a tempting but incorrect answer is \([-2,3]\). That answer includes \(x=-2\), even though there is no graph point with that horizontal coordinate.
The correction is to read the markers, not just the extreme horizontal positions: \(-2\) is excluded because its endpoint is open, and \(3\) is included because its endpoint is closed. More generally, an endpoint is included only if the graph actually contains a point with that horizontal coordinate.
A Reliable Procedure
Use the following sequence when reading a domain from a function graph. It keeps the focus on horizontal coordinates and makes endpoint decisions explicit.
Separate branches, segments, and isolated points before combining their contributions.
Record the inputs above which that piece has at least one plotted point.
Use open and closed markers to decide inclusion, and use arrows to identify continuation beyond the visible portion.
Do not include horizontal positions that have no point on any piece of the graph.
Combine the intervals and isolated inputs, then confirm that each included and excluded boundary agrees with the graph.
Why Careful Reading Matters
A graph can make the general shape of a function easy to see, but a domain answer depends on details that may be visually small. An open circle and a closed dot can be close together, and a short gap can be easy to overlook. Arrows can also be mistaken for ordinary endpoints if their direction is not checked.
Use a simple test for each candidate input: Is there a plotted point directly above this horizontal position? If yes, the input belongs to the domain; if not, it does not. This test handles continuous branches, isolated points, gaps, and endpoints consistently. It also keeps the task distinct from reading the range, which concerns vertical rather than horizontal coordinates.
Check Your Understanding
For each question, base your answer on the horizontal coordinates of the plotted points.
- A curve segment runs from an open endpoint at \(x=-5\) to a closed endpoint at \(x=2\), including every point between them. What is its domain in interval notation?
- A branch has an arrow pointing to the left and ends at a closed point with horizontal coordinate \(x=4\). What set of inputs does this branch contribute?
- A graph has an open endpoint at \(x=1\), but also has a separate closed point whose horizontal coordinate is \(1\). Is \(1\) in the domain? Explain.
- A graph has one piece projecting onto \([-3,-1]\) and another projecting onto \([2,\infty)\). Write the domain and identify the gap.
- Why does an arrow pointing right not represent a single included endpoint at the edge of the displayed graph?
