Introduction
A graph displays a collection of points, and each point has both a horizontal coordinate and a vertical coordinate. When we read the range, we collect the vertical coordinates that actually occur. The horizontal positions help locate the points, but the range is about their heights.
A useful way to picture the task is to project the graph onto the vertical axis: imagine moving every plotted point straight across to the \(y\)-axis and recording all the heights reached. The result may be one interval, several separated pieces, or a set that continues without bound.
Range as the Set of Vertical Coordinates
For example, if the graph contains the point \((2,5)\), then the output value \(5\) occurs. The range records \(5\), not the input \(2\). If several different points have the same height, that height is still listed only once in the range.
To read the range, scan the graph vertically rather than from left to right. Locate the lowest and highest heights reached, then check whether every height between them occurs. Also check whether any endpoint height is excluded and whether the graph continues beyond the visible interval.
Projecting a Graph onto the Vertical Axis
The horizontal-line test and the projection method express the same idea. Each point on the graph has a vertical coordinate. The projection collects those coordinates; a horizontal line at height \(y\) intersects the graph precisely when at least one graph point has vertical coordinate \(y\).
This gives a practical method:
Look for the lowest and highest points, as well as any arrows that show continuation.
Ask whether the graph reaches every intermediate height or whether there are gaps.
A filled point is included; an open circle marks a point that is not part of the graph.
Use interval notation for continuous stretches and separate pieces when some heights are missing.
These steps concern the values actually represented by the graph. Do not include a height just because the graph approaches it, or omit a height just because it occurs at more than one horizontal position.
Worked Example: A Segment with Two Included Endpoints
Worked Example: A Segment with Two Included Endpoints
Suppose a graph is the straight line segment joining the filled points \((-2,-1)\) and \((3,4)\). What is its range?
The lowest point is the left endpoint, whose vertical coordinate is \(-1\). The highest point is the right endpoint, whose vertical coordinate is \(4\). Both points are filled, so both endpoint heights are included.
The segment contains every point between its endpoints. In particular, as the graph runs from height \(-1\) to height \(4\), it reaches every intermediate height. Therefore the range is
Check: the bottom height \(-1\) occurs at \((-2,-1)\), and the top height \(4\) occurs at \((3,4)\). Since the drawn segment connects those points, every height between them occurs as well. No height below \(-1\) or above \(4\) is on the segment.
Open and Closed Endpoints
Endpoint symbols determine whether a boundary value belongs to the range, but only when that height is attained nowhere else on the graph. A filled point is part of the graph, so its height is included. An open circle is not part of the graph, so that particular point does not contribute its height.
Suppose the graph rises continuously from an open endpoint at height \(-2\) and has an arrow showing that it keeps rising without bound. It reaches every height greater than \(-2\), but it never reaches \(-2\) at the open endpoint. Its range is \((-2,\infty)\), not \([-2,\infty)\).
Endpoint symbols must be read in context. If another part of the graph contains a filled point at the same height as an open endpoint, that height is still in the range. What matters is whether any graph point has that vertical coordinate.
Worked Example: An Open Endpoint and an Upward Arrow
Worked Example: An Open Endpoint and an Upward Arrow
Imagine a continuous curve that begins at an open circle at \((1,-2)\), rises as it moves to the right, and has an arrow showing that it continues upward without bound. Read its range.
The curve starts just above height \(-2\). Because the starting point is open, the graph does not contain the point \((1,-2)\). The height \(-2\) is therefore not included. As the curve rises continuously, it reaches each height greater than \(-2\). The arrow shows that there is no highest height.
Thus the range is
Check: every \(y>-2\) is reached along the continuous rising curve, while \(y=-2\) is excluded and no value below \(-2\) is reached. The upward arrow accounts for arbitrarily large values.
Arrows, Gaps, and Separate Pieces
An arrow shows that the graph continues in the direction indicated. To use it for the range, inspect the arrow’s vertical movement. An arrow that rises without bound means the range has no upper bound. An arrow that descends without bound means the range has no lower bound. A graph can also extend indefinitely to the left or right while staying within a limited band of heights; horizontal continuation alone does not imply an unbounded range.
A graph may contain gaps in its set of heights. A gap occurs when some horizontal line does not intersect any part of the graph. In that case the range must leave out the corresponding value or interval, even if the graph has points both below and above the gap.
Disconnected pieces do not automatically produce a gap in the range. Two separated pieces might reach overlapping heights, or one piece might fill heights left out by another. Project all the pieces onto the vertical axis and combine the heights they reach.
Worked Example: Two Separate Height Intervals
Suppose a graph has one continuous piece that reaches every height from \(0\) up to, but not including, \(1\). A second piece reaches every height greater than \(1\) and up to and including \(2\). Neither piece contains a point at height \(1\). What is the range?
The first piece contributes \([0,1)\): height \(0\) is included, while \(1\) is not. The second contributes \((1,2]\): \(1\) is excluded, while \(2\) is included. Combining the pieces gives
Check: every height from \(0\) through \(2\) occurs except \(1\). A horizontal line at height \(1\) misses both pieces, so \(1\) must not be included. The filled lower and upper endpoint heights, \(0\) and \(2\), are included.
Why Projection Gives the Range
The projection method works directly from the definition. Let \(G\) be the set of points on the graph of \(f\). Each point in \(G\) has the form \((x,f(x))\), where \(x\) is an allowed input. Projecting a point onto the vertical axis keeps its vertical coordinate and discards its horizontal coordinate.
This argument also explains why repeated outputs appear only once in the range: the range is a set of vertical coordinates, not a count of how many graph points have each height.
A Common Mistake: Reading the Horizontal Extent
Worked Example: Correcting a Range Reading
Suppose a graph is a horizontal segment from the filled point \((-3,2)\) to the filled point \((4,2)\). A common incorrect reading is to report the range as \([-3,4]\), using the segment’s horizontal coordinates.
That interval describes the horizontal extent, not the heights. Every point on the segment has vertical coordinate \(2\). For instance, the endpoints have coordinates \((-3,2)\) and \((4,2)\), and points between them also have height \(2\). There are no graph points at any other height.
Therefore the range is the single-element set
Check: the horizontal line \(y=2\) meets the graph, and every horizontal line at a different height misses it. Thus \(2\) is the only output value.
The same distinction matters for curves that move back and forth horizontally. A graph may extend across a large interval of \(x\)-values while reaching only a small collection of \(y\)-values. Always read the vertical coordinates when finding range.
Why Reading Range Matters
The range tells which outputs a function can actually produce. It helps distinguish a value that is possible from one that is merely suggested by the graph’s overall shape. Correctly identifying included endpoints and gaps is essential: a single omitted height changes the set of outputs.
When reading a graph, keep three questions separate:
- What are the lowest and highest heights reached?
- Are those boundary heights included, and does the graph continue beyond them?
- Does every height between the boundaries occur, or are some heights missing?
Answering these questions before writing interval notation reduces errors. A final horizontal-line check is useful: each value you include should correspond to a horizontal line that meets the graph, and each value you exclude should correspond to one that does not.
Check Your Understanding
For each question, focus on the graph’s vertical coordinates rather than its horizontal extent.
- A segment joins the filled points \((0,-3)\) and \((2,1)\). What is its range?
- A continuous curve begins at an open circle of height \(4\) and continues downward without bound. What is its range?
- What does an upward arrow tell you about the range if it indicates that the graph rises without bound?
- A graph has an open circle at height \(2\), but another filled point elsewhere also has height \(2\). Is \(2\) in the range? Explain.
- Two graph pieces reach \([1,3]\) and \((3,5]\). What is their combined range?
