The Central Idea
The domain and range of a graph can be read by asking two different questions.
- Domain: Which horizontal positions contain at least one point of the graph?
- Range: Which vertical positions contain at least one point of the graph?
In set language,
\[ \boxed{\operatorname{Domain}=\{x:\text{the graph contains at least one point }(x,y)\}}, \] \[ \boxed{\operatorname{Range}=\{y:\text{the graph contains at least one point }(x,y)\}}. \]Domain Is About Inputs
For a function written as
\[ y=f(x), \]the domain consists of the input values for which the function produces an output.
Graphically, an input value belongs to the domain if a vertical line through that value reaches at least one point on the graph.
Range Is About Outputs
The range consists of the output values that actually occur.
Graphically, an output value belongs to the range if a horizontal line through that value reaches at least one point on the graph.
The graph therefore tells us not merely what inputs are permitted, but what outputs are realized.
Read Domain Horizontally
To read the domain, imagine projecting the graph onto the \(x\)-axis.
Every \(x\)-value covered by that horizontal projection belongs to the domain.
A useful mental question is:
How far left and right does the graph exist?
Read Range Vertically
To read the range, imagine projecting the graph onto the \(y\)-axis.
Every \(y\)-value covered by that vertical projection belongs to the range.
A useful mental question is:
How far down and up does the graph exist?
Domain and Range Belong to Relations Too
Domain and range are not limited to functions.
Any relation made of ordered pairs has a domain and a range. If
\[ R=\{(x,y)\}, \]then the domain is the set of first coordinates and the range is the set of second coordinates.
A graph may fail the vertical-line test and still have a perfectly valid domain and range.
Example: A Finite Set of Points
Suppose a graph contains only the points
\[ (-3,2),\quad(0,5),\quad(4,2),\quad(7,-1). \]The domain is the set of distinct \(x\)-coordinates:
\[ \boxed{\{-3,0,4,7\}}. \]The range is the set of distinct \(y\)-coordinates:
\[ \boxed{\{-1,2,5\}}. \]The repeated output \(2\) is listed only once because a set does not repeat elements.
Discrete Graphs Use Individual Values
When a graph consists of isolated points, domain and range are usually best written as sets of individual values rather than as intervals.
For example,
\[ \{1,2,3,4\} \]does not mean the same thing as
\[ [1,4]. \]The interval includes every real number between \(1\) and \(4\); the finite set includes only four values.
Continuous Graphs Often Produce Intervals
If a graph exists continuously across all horizontal positions from \(x=-2\) through \(x=5\), including both endpoints, then its domain is
\[ \boxed{[-2,5]}. \]If its outputs fill every value from \(y=-1\) through \(y=7\), including both extremes, then its range is
\[ \boxed{[-1,7]}. \]Closed Endpoints Are Included
A filled or closed endpoint indicates that the endpoint itself belongs to the graph.
If a graph begins at the closed point
\[ (-4,1) \]and continues to the right, then \(x=-4\) is included in the domain.
The domain might therefore begin with
\[ [-4,\ldots). \]Open Endpoints Are Excluded
An open circle indicates that the point shown is not part of the graph.
If a graph begins at an open point with \(x=-4\) and then continues to the right, the domain begins with
\[ (-4,\ldots). \]Parentheses record that \(-4\) is excluded.
An Open Point Does Not Automatically Remove an Input
A hole at one point removes that ordered pair, but the same \(x\)-value might appear elsewhere on the graph.
Suppose there is an open circle at
\[ (2,3) \]but a filled point at
\[ (2,-1). \]Then \(x=2\) still belongs to the domain because the graph contains the point \((2,-1)\).
An Open Point Does Not Automatically Remove an Output
The same reasoning applies to range.
If \((2,3)\) is open but some other included point has \(y=3\), then \(y=3\) remains in the range.
To determine whether a value is excluded, inspect the entire graph—not only one point.
Arrows Indicate Continued Behavior
An arrow on a graph indicates that the graph continues beyond the portion drawn.
If a curve extends indefinitely to the left and right, then its domain may be
\[ \boxed{(-\infty,\infty)}. \]The visible edge of the coordinate grid is not an endpoint when an arrow shows continuation.
Infinity Is Not an Endpoint
The symbols \(\infty\) and \(-\infty\) describe unbounded behavior. They are not real numbers that can be included as endpoints.
Therefore, interval notation always uses parentheses with infinity:
\[ (-\infty,5], \]never
\[ [-\infty,5]. \]Example: A Line
A nonvertical line such as
\[ y=2x-3 \]extends indefinitely in both horizontal directions and both vertical directions.
Therefore,
\[ \boxed{\operatorname{Domain}=(-\infty,\infty)}, \] \[ \boxed{\operatorname{Range}=(-\infty,\infty)}. \]Example: A Horizontal Line
Consider
\[ y=4. \]The graph extends through every real \(x\)-value, so
\[ \operatorname{Domain}=(-\infty,\infty). \]But every point has the same output \(4\), so
\[ \boxed{\operatorname{Range}=\{4\}}. \]Example: A Vertical Line
Consider the relation
\[ x=-2. \]Every point has the same first coordinate, so
\[ \boxed{\operatorname{Domain}=\{-2\}}. \]The line extends through every real \(y\)-value, so
\[ \boxed{\operatorname{Range}=(-\infty,\infty)}. \]This graph is a relation but not a function of \(x\).
Example: A Ray
Suppose a ray begins at the closed point \((-1,2)\) and extends upward and to the right indefinitely.
Its domain is
\[ \boxed{[-1,\infty)}, \]and its range is
\[ \boxed{[2,\infty)}. \]Example: A Line Segment
Suppose a segment joins the filled endpoints
\[ (-3,4) \quad\text{and}\quad (5,-2). \]Because every horizontal position between \(-3\) and \(5\) appears,
\[ \boxed{\operatorname{Domain}=[-3,5]}. \]Because every vertical position between \(-2\) and \(4\) appears,
\[ \boxed{\operatorname{Range}=[-2,4]}. \]The Leftmost and Rightmost Points
For a bounded graph, the leftmost and rightmost included horizontal positions often determine the endpoints of the domain.
Before writing the interval, check whether those positions are included, excluded, or merely the edge of the viewing window.
The Lowest and Highest Points
For a bounded graph, the lowest and highest included vertical positions often determine the endpoints of the range.
These values may occur at endpoints, turning points, isolated points, or elsewhere on the graph.
Domain Is Not Found by Looking Only at Endpoints
Endpoints tell us where a graph may begin or end, but gaps inside the graph can split the domain.
A graph existing for
\[ -5\le x<-1 \]and again for
\[ 2\le x\le6 \]has domain
\[ \boxed{[-5,-1)\cup[2,6]}. \]Disconnected Graphs Produce Unions
When separate pieces occupy separate input intervals, interval notation uses the union symbol \(\cup\).
For example,
\[ (-\infty,-3)\cup(1,\infty) \]means all real values less than \(-3\) together with all real values greater than \(1\).
Range Can Also Be Disconnected
Separate graphical pieces can leave missing output values as well.
If one portion of a graph produces
\[ y\le-2 \]and another produces
\[ y>3, \]the range is
\[ \boxed{(-\infty,-2]\cup(3,\infty)}. \]Holes Can Split an Interval
Suppose a curve exists for every \(x\) from \(-4\) to \(6\) except \(x=2\), and no other point on the graph has first coordinate \(2\).
Then the domain is
\[ \boxed{[-4,2)\cup(2,6]}. \]The single missing input breaks the interval into two pieces.
Set-Builder Notation Can Describe the Same Domain
The domain
\[ (-\infty,2)\cup(2,\infty) \]can also be written
\[ \boxed{\{x\in\mathbb{R}:x\ne2\}}. \]Interval notation emphasizes intervals; set-builder notation emphasizes the defining condition.
Graphs Can Mix Intervals and Isolated Values
Suppose a graph contains a curve for
\[ -2\le x<4 \]and also an isolated point whose \(x\)-coordinate is \(7\).
The domain is
\[ \boxed{[-2,4)\cup\{7\}}. \]Not every domain can be expressed using intervals alone.
Piecewise Graphs Require Whole-Graph Reading
A piecewise-defined function may contain several formulas, but domain and range belong to the complete graph.
Read all pieces together, combine overlapping input intervals, and combine overlapping output intervals before writing the final sets.
Example: Two Piecewise Segments
Suppose one segment covers
\[ -4\le x\le0 \]and a second covers
\[ 0The fact that the pieces use different endpoint conventions at \(x=0\) does not create a gap because at least one included point exists there.
Do Not Double-Count Overlap
Suppose one piece has domain \([-3,2]\) and another has domain \([1,6]\).
Their union is simply
\[ \boxed{[-3,6]}, \]because the intervals overlap from \(1\) through \(2\).
A Parabola Illustrates Unbounded Range
Consider the graph of
\[ y=x^2. \]Every real input is allowed, so
\[ \operatorname{Domain}=(-\infty,\infty). \]The graph has a lowest output of \(0\) and rises without bound, so
\[ \boxed{\operatorname{Range}=[0,\infty)}. \]Vertical Shifts Change Range
For
\[ y=x^2-5, \]the domain remains all real numbers, but the lowest output becomes \(-5\).
\[ \boxed{\operatorname{Range}=[-5,\infty)}. \]A transformation can change domain, range, both, or neither depending on how the graph moves.
A Square-Root Graph Illustrates a Restricted Domain
Consider
\[ y=\sqrt{x}. \]The graph begins at \((0,0)\) and extends to the right and upward.
Therefore,
\[ \boxed{\operatorname{Domain}=[0,\infty)}, \] \[ \boxed{\operatorname{Range}=[0,\infty)}. \]A Shifted Square-Root Graph
If a square-root graph begins at the filled point \((3,-2)\) and extends rightward and upward, then
\[ \boxed{\operatorname{Domain}=[3,\infty)}, \] \[ \boxed{\operatorname{Range}=[-2,\infty)}. \]The starting point controls both lower bounds in this example.
An Absolute-Value Graph
For
\[ y=|x|, \]the graph extends indefinitely left and right, so its domain is all real numbers.
Its outputs never fall below \(0\), so
\[ \boxed{\operatorname{Range}=[0,\infty)}. \]A Circle Shows Why Relations Matter
Consider the circle
\[ x^2+y^2=25. \]The leftmost and rightmost points occur at \(x=-5\) and \(x=5\), so
\[ \boxed{\operatorname{Domain}=[-5,5]}. \]The lowest and highest points occur at \(y=-5\) and \(y=5\), so
\[ \boxed{\operatorname{Range}=[-5,5]}. \]The circle is not a function of \(x\), but domain and range are still defined normally.
An Ellipse Can Be Read the Same Way
Suppose an ellipse extends horizontally from \(x=-6\) to \(x=6\) and vertically from \(y=-2\) to \(y=2\).
Then
\[ \boxed{\operatorname{Domain}=[-6,6]}, \] \[ \boxed{\operatorname{Range}=[-2,2]}. \]The horizontal and vertical extents need not be equal.
A Reciprocal Graph Has a Missing Input
Consider
\[ y=\frac{1}{x}. \]The graph has no point with \(x=0\), so
\[ \boxed{\operatorname{Domain}=(-\infty,0)\cup(0,\infty)}. \]It also never produces \(y=0\), so
\[ \boxed{\operatorname{Range}=(-\infty,0)\cup(0,\infty)}. \]Missing Values Must Be Verified Across the Whole Graph
A vertical gap near \(x=2\) may suggest that \(2\) is excluded from the domain, but that conclusion is valid only if no other point on the graph has first coordinate \(2\).
Similarly, a horizontal gap near \(y=5\) removes \(5\) from the range only if no included point anywhere on the graph has output \(5\).
A Vertical Asymptote Usually Signals a Domain Exclusion
If a graph approaches the vertical line
\[ x=a \]without containing any point whose first coordinate is \(a\), then \(x=a\) is excluded from the domain.
The graphical evidence is the absence of points at that input—not merely the presence of the asymptote.
A Horizontal Asymptote Does Not Automatically Remove a Range Value
A graph may approach a horizontal line as \(x\) moves far away and still cross that same horizontal level somewhere else.
Therefore, a horizontal asymptote alone does not prove that its \(y\)-coordinate is excluded from the range.
Inspect the entire graph before excluding the value.
Exponential Graphs Often Have All Real Inputs
For the basic exponential function
\[ y=2^x, \]every real input is allowed:
\[ \operatorname{Domain}=(-\infty,\infty). \]The outputs are always positive:
\[ \boxed{\operatorname{Range}=(0,\infty)}. \]Logarithmic Graphs Reverse the Pattern
For
\[ y=\log_2 x, \]only positive inputs occur, so
\[ \boxed{\operatorname{Domain}=(0,\infty)}. \]The outputs extend without bound in both directions, so
\[ \boxed{\operatorname{Range}=(-\infty,\infty)}. \]Restrictions Can Change a Familiar Graph
Suppose the graph of
\[ y=x^2 \]is restricted to
\[ -2\le x\le3. \]The domain is immediately
\[ \boxed{[-2,3]}. \]The range must then be read from only that displayed portion of the parabola, not from the unrestricted function.
A Formula's Usual Domain Is Not Always the Graphed Domain
A familiar formula may be shown only on a restricted interval.
If the graph displays \(y=x^2\) only for \(-1\le x\le4\), then the graph's domain is \([-1,4]\), even though the unrestricted formula \(x^2\) is defined for every real \(x\).
Read the graph actually given.
The Graphing Window Is Not a Restriction Unless the Graph Says So
A coordinate screen might display only
\[ -10\le x\le10. \]That does not mean the graph's domain is \([-10,10]\).
If the curve continues beyond the screen or the formula is understood to continue, the actual domain can be much larger.
Clipped Curves Require Caution
A curve that disappears at the top, bottom, left, or right edge of a graphing window may simply be leaving the window.
Do not treat the clipping point as an endpoint unless the graph marks an endpoint explicitly or the surrounding information establishes one.
Arrows Resolve Window Ambiguity
Hand-drawn textbook graphs often use arrows to show that a curve continues.
When arrows are present, use them as direct evidence of unbounded or continuing behavior rather than treating the frame as a boundary.
Read the Axis Scale Before Reading Values
Domain and range values can be misread when tick marks do not represent one unit each.
If the \(x\)-axis is labeled in increments of \(5\), the third tick to the right of zero represents \(15\), not \(3\).
Scale is part of the graph's mathematical information.
Unequal Axis Scales Do Not Change the Definitions
The horizontal axis might use increments of \(2\) while the vertical axis uses increments of \(10\).
Domain still comes from horizontal coordinates and range still comes from vertical coordinates. Only the numerical reading of those coordinates changes.
Estimate When an Endpoint Falls Between Tick Marks
A graph may indicate an endpoint near \(x=3.4\) rather than at an exact labeled tick.
If the coordinate is being read graphically, report an appropriate estimate:
\[ x\approx3.4. \]Do not imply more precision than the graph supports.
Exact and Estimated Domains Can Look Different
An algebraic description might establish the exact endpoint
\[ x=\sqrt{2}, \]while a graph suggests only
\[ x\approx1.41. \]When exact information is available, retain the exact value. When the graph is the only source, use an estimate consistent with its resolution.
Domain from a Table and Domain from a Graph Agree
If a graph was created from the discrete table
| \(x\) | \(y\) |
|---|---|
| \(-2\) | \(4\) |
| \(0\) | \(1\) |
| \(3\) | \(1\) |
| \(5\) | \(-2\) |
then
\[ \operatorname{Domain}=\{-2,0,3,5\} \]and
\[ \operatorname{Range}=\{-2,1,4\}. \]Connecting Discrete Points Changes the Relation
If four isolated data points are connected by line segments, the resulting graph contains many additional points.
Consequently, its domain and range may become intervals rather than finite sets.
Connect points only when the context or definition justifies values between the observations.
Context Can Restrict Domain
A mathematical formula may allow more inputs than a real-world model does.
If \(t\) represents time elapsed after an experiment begins, negative values of \(t\) may be irrelevant even if the underlying algebraic formula accepts them.
A graph of the model should reflect the intended contextual domain.
Context Can Restrict Range
If a graph represents a number of objects, only outputs meaningful in that context may be allowed.
A continuous mathematical curve might suggest fractional or negative outputs that the modeled quantity cannot actually take.
Always distinguish the pure mathematical relation from the modeled interpretation when context is supplied.
Domain and Range of an Inequality Region
A shaded inequality region also has a domain and range as a relation.
Ask which \(x\)-coordinates occur anywhere in the shaded region and which \(y\)-coordinates occur anywhere in that region.
Because a region often extends broadly, its domain or range may be all real numbers even when its boundary is restricted.
Example: A Half-Plane
Consider the inequality
\[ y\ge2x-1. \]For every real \(x\), there are shaded points above the boundary line, so
\[ \operatorname{Domain}=(-\infty,\infty). \]Likewise, for every real \(y\), some sufficiently chosen \(x\) produces a point in the region. Thus
\[ \boxed{\operatorname{Range}=(-\infty,\infty)}. \]Boundaries and Regions Have Different Domains and Ranges
The boundary curve of an inequality and the full shaded solution region are different relations.
When asked for domain or range, determine whether the question refers to the boundary alone or to the complete shaded set.
Systems Can Restrict the Visible Relation
A system of inequalities may produce a bounded feasible region such as a polygon.
The domain of that region is determined by its leftmost and rightmost included \(x\)-values, while the range is determined by its lowest and highest included \(y\)-values.
Interior points count just as much as boundary points.
Domain Can Be Read with Vertical Sweeps
Imagine moving a vertical test line from left to right.
Whenever the line touches at least one point of the graph, that horizontal position belongs to the domain.
This is not the vertical-line test for functions. Here the question is only whether the vertical line touches the relation at least once.
Range Can Be Read with Horizontal Sweeps
Imagine moving a horizontal line from bottom to top.
Whenever the line touches at least one point of the graph, that vertical position belongs to the range.
This method works for functions and nonfunction relations alike.
A Function Can Repeat Outputs
A function may have several inputs that produce the same output.
For example, on
\[ y=x^2, \]both \(x=2\) and \(x=-2\) produce \(y=4\).
The range records the output value \(4\) once. It does not record how many inputs produced it.
Range Is Not the Same as the Set of y-Intercepts
The \(y\)-intercept tells us only the output associated with \(x=0\).
Range asks for every output represented anywhere on the graph.
A graph may have one \(y\)-intercept and infinitely many range values.
Domain Is Not the Same as the Set of x-Intercepts
An \(x\)-intercept is an input where the graph has output \(0\).
Domain asks for every input represented on the graph, whether or not the output is zero.
Intercepts are special points; domain and range describe whole coordinate sets.
A Graph Can Have Empty Domain and Range
If a relation has no points at all, it is the empty relation.
Its domain and range are both empty:
\[ \boxed{\operatorname{Domain}=\varnothing}, \] \[ \boxed{\operatorname{Range}=\varnothing}. \]A Single Point Has One-Element Domain and Range
For the graph containing only
\[ (4,-3), \]the domain and range are
\[ \boxed{\operatorname{Domain}=\{4\}}, \] \[ \boxed{\operatorname{Range}=\{-3\}}. \]Endpoint Inclusion Can Differ Between Domain and Range
Consider a curve beginning with an open point at \((-2,1)\) and ending with a filled point at \((4,5)\).
The input \(-2\) may be excluded while the output \(1\) could still belong to the range if the curve reaches \(y=1\) somewhere else.
Endpoint symbols must therefore be interpreted separately for horizontal and vertical coordinate sets.
Range Requires Watching the Entire Curve
A graph can rise, fall, loop, or return to previous output levels.
Do not infer the range merely from the \(y\)-coordinates of its leftmost and rightmost points. The highest or lowest output may occur in the interior.
Domain Requires Watching the Entire Relation
For a loop, circle, sideways parabola, or other nonfunction relation, the leftmost and rightmost points may occur far from the graph's visually highest or lowest parts.
Read horizontal extent independently of vertical extent.
Example: A Sideways Parabola
Suppose a relation has the form
\[ x=y^2-4. \]Its leftmost point occurs at \((-4,0)\), and the graph extends indefinitely to the right. Thus
\[ \boxed{\operatorname{Domain}=[-4,\infty)}. \]Every real \(y\)-value occurs, so
\[ \boxed{\operatorname{Range}=(-\infty,\infty)}. \]Domain and Range Can Be Symmetric
Some graphs have the same domain and range, such as the circle
\[ x^2+y^2=25. \]Others do not. Similarity between domain and range is a property of the particular graph, not a general rule.
Use the Most Natural Notation
Different kinds of sets are easiest to communicate in different ways.
- Use interval notation for continuous intervals of real values.
- Use set notation for isolated values.
- Use unions for disconnected pieces.
- Use set-builder notation when a condition is more concise than a long union.
Interval Notation Review
| Condition | Interval |
|---|---|
| \(a\le x\le b\) | \([a,b]\) |
\(a| \((a,b)\) | |
\(a\le x| \([a,b)\) | |
| \(x>a\) | \((a,\infty)\) |
| \(x\le b\) | \((-\infty,b]\) |
Brackets Mean Inclusion
A square bracket indicates that a finite endpoint belongs to the set.
\[ [2,7] \]includes both \(2\) and \(7\).
Parentheses Mean Exclusion
A parenthesis indicates that a finite endpoint does not belong to the set.
\[ (2,7) \]contains values strictly between \(2\) and \(7\), but not the endpoints.
Union Means Either Piece
The notation
\[ [-5,-1)\cup(2,8] \]means a value belongs to the set if it lies in the first interval or the second interval.
The gap from \(-1\) through \(2\) is not automatically included.
Graphical Domain Can Reveal Algebraic Restrictions
A missing vertical slice in a graph often corresponds to an algebraic input restriction such as division by zero, an even root of a negative quantity, or a logarithm of a nonpositive quantity.
The graph lets us observe the restriction before or without deriving it algebraically.
Graphical Range Can Reveal Output Restrictions
A graph may remain entirely above a horizontal line, entirely below one, or between two horizontal levels.
Those visible restrictions become range conditions.
For example, if the graph never falls below \(y=-3\) and reaches \(-3\), the range has lower bound \(-3\) included.
Do Not Assume Every Value Between Two Outputs Occurs
A disconnected graph may contain points at \(y=-4\) and \(y=4\) without containing any points at \(y=0\).
The presence of a low value and a high value does not guarantee every value between them unless the graphical pieces actually fill the interval.
Continuity Matters When Filling Intervals
If a connected continuous curve moves from an output of \(-2\) to an output of \(5\), the curve passes through intermediate output levels.
For disconnected or discrete graphs, that conclusion may fail.
The next lessons will develop graphical behavior more systematically; here the important task is to read what is actually present.
Domain and Range from a Graph Can Be Approximate
When an unlabeled graph is drawn from measured data, the extreme coordinates may not be exact mathematical constants.
A reasonable report might be
\[ \operatorname{Domain}\approx[1.2,8.7]. \]The approximation symbol communicates the limits of graphical measurement.
Technology Can Help Inspect Domain and Range
Graphing software can zoom, trace, and evaluate points, but it does not remove the need to interpret the graph correctly.
A default viewing window can hide holes, branches, endpoints, or additional behavior. Use technology to inspect the relation, not as a substitute for understanding it.
Zooming Can Reveal Missing Values
A hole or narrow gap may be visually invisible at one scale.
Zooming can reveal whether a point is filled, open, disconnected, or merely drawn too small to distinguish clearly.
Zooming Out Can Reveal Continued Behavior
A graph that appears bounded in a small window may continue well beyond it.
Zooming out can help determine whether a visible extreme is a true endpoint, a turning point, or simply a temporary feature of the selected window.
Graphical Evidence Should Match Symbolic Evidence
When both a formula and a graph are given, compare them.
If the formula is
\[ y=\frac{1}{x-3}, \]then \(x=3\) is algebraically excluded. A graph showing a point at \(x=3\) would signal an error in plotting or interpretation.
Use Substitution to Check a Questionable Input
If the graph appears to omit \(x=4\), and a formula is available, substitute \(x=4\).
If the expression is defined and produces an output, then the input belongs to the formula's domain even if the plotted image is unclear.
Graphical and algebraic representations should ultimately agree when they describe the same relation with the same restrictions.
Use an Equation to Check a Questionable Output
Determining whether a particular \(y\)-value belongs to the range can be reframed as asking whether the graph intersects the horizontal line
\[ y=c. \]Algebraically, this becomes asking whether the relation has at least one real solution with output \(c\).
Range Connects Naturally to Horizontal Intersections
A number \(c\) belongs to the range exactly when the horizontal line
\[ y=c \]intersects the graph at least once.
This turns range membership into an intersection question.
Domain Connects Naturally to Vertical Intersections
A number \(a\) belongs to the domain exactly when the vertical line
\[ x=a \]intersects the graph at least once.
The number of intersections is irrelevant to domain membership; one or more is enough.
The Vertical-Line Test Has a Different Purpose
The vertical-line test asks whether a relation is a function by checking whether any vertical line meets the graph more than once.
Domain asks only whether the vertical line meets the graph at all.
These tests use similar visual tools but answer different questions.
Example: Domain of a Nonfunction Relation
A circle may be intersected twice by many vertical lines, so it is not a function of \(x\).
Yet every vertical line between its leftmost and rightmost points intersects it at least once, so those \(x\)-values form its domain.
Common Error: Reversing Domain and Range
Domain corresponds to horizontal coordinates, and range corresponds to vertical coordinates.
A useful reminder is
\[ \boxed{\text{domain}\leftrightarrow x,\qquad\text{range}\leftrightarrow y}. \]Common Error: Treating the Window as the Domain
If the screen displays \(-10\le x\le10\), it is incorrect to report that interval as the domain merely because those are the visible horizontal limits.
Determine whether the graph actually stops there.
Common Error: Including an Open Endpoint
An open circle at \(x=5\) means that particular endpoint is excluded.
If no other point on the graph uses \(x=5\), the interval should use a parenthesis at \(5\), not a bracket.
Common Error: Excluding a Coordinate That Appears Elsewhere
A hole at \((2,4)\) does not remove \(x=2\) from the domain if an included point such as \((2,-3)\) also exists.
Likewise, it does not remove \(y=4\) from the range if another included point has output \(4\).
Common Error: Using Brackets with Infinity
Infinity is never included as an endpoint.
Write
\[ [3,\infty), \]not
\[ [3,\infty]. \]Common Error: Turning Discrete Values into an Interval
If a graph contains points only at \(x=1,3,5\), its domain is
\[ \boxed{\{1,3,5\}}, \]not \([1,5]\).
The interval would incorrectly include inputs such as \(2\) and \(4\).
Common Error: Assuming a Hole Always Removes a Range Value
A hole at height \(y=6\) removes only that point.
If another included point lies at height \(6\), then \(6\) remains in the range.
Range is determined by all points at that horizontal level.
Common Error: Reading Only the Formula
A formula may be graphed under an explicit restriction.
If the displayed relation is \(y=x^2\) for \(-3\le x\le2\), using the unrestricted polynomial domain would ignore the graph actually under study.
Common Error: Reading Only the Endpoints for Range
A curve can rise above both endpoint heights or fall below both endpoint heights before returning.
Scan the entire graph vertically before deciding the range.
Common Error: Assuming Every Intermediate Output Occurs
On a disconnected or discrete graph, outputs can jump over entire intervals.
Confirm that the graph actually occupies the intermediate horizontal levels before including them in the range.
A Reliable Procedure for Finding Domain from a Graph
- Read the horizontal axis and its scale.
- Identify every horizontal position where at least one graph point exists.
- Locate left and right endpoints, if any.
- Check whether each finite endpoint is open or closed.
- Look for holes, gaps, isolated points, and separate pieces.
- Interpret arrows as continuation rather than endpoints.
- Distinguish the graph's actual extent from the viewing window.
- Combine the result using intervals, unions, or discrete sets.
- Verify questionable values against the formula or context when available.
A Reliable Procedure for Finding Range from a Graph
- Read the vertical axis and its scale.
- Identify every vertical position where at least one graph point exists.
- Locate the lowest and highest outputs, if they exist.
- Check whether boundary outputs are actually attained.
- Look for missing output levels, holes, isolated points, and separate pieces.
- Interpret arrows as continuation rather than boundaries.
- Do not infer range from the visible window alone.
- Combine intervals and isolated values as needed.
- Verify uncertain output values with horizontal-line intersections or algebra.
A Fast Visual Method
For domain, mentally cast the graph's shadow onto the \(x\)-axis.
For range, mentally cast its shadow onto the \(y\)-axis.
Then inspect that shadow for open endpoints, gaps, isolated values, and unbounded continuation.
Example: Reading a Bounded Continuous Graph
Suppose a continuous curve begins at an open point with \(x=-3\), ends at a filled point with \(x=5\), reaches a lowest included output of \(-4\), and a highest included output of \(6\).
If no internal gaps occur, then
\[ \boxed{\operatorname{Domain}=(-3,5]}, \] \[ \boxed{\operatorname{Range}=[-4,6]}. \]Example: Reading a Graph with a Hole
Suppose a curve otherwise covers every input from \(-2\) through \(7\), but the only point at \(x=3\) is open.
Then
\[ \boxed{\operatorname{Domain}=[-2,3)\cup(3,7]}. \]Whether the hole's \(y\)-coordinate is removed from the range depends on whether that same output occurs anywhere else.
Example: Reading Two Disconnected Pieces
Suppose one graph piece occupies inputs \(-6\le x<-1\), while another
occupies inputs \(2
The domain is
The gap between \(-1\) and \(2\) contains no represented inputs.
Example: Discrete Inputs with Repeated Outputs
Suppose a graph contains
\[ (-2,1),\quad(1,4),\quad(3,1),\quad(6,7). \]Then
\[ \boxed{\operatorname{Domain}=\{-2,1,3,6\}}, \] \[ \boxed{\operatorname{Range}=\{1,4,7\}}. \]Example: A Restricted Parabola
Consider
\[ y=x^2, \qquad -2\le x<3. \]The domain is
\[ \boxed{[-2,3)}. \]The smallest output is \(0\), attained at \(x=0\). The output approaches \(9\) as \(x\to3^-\), but \(9\) is not attained at the excluded endpoint. However, \(y=9\) would require \(x=\pm3\), neither of which is in the restricted domain. Therefore,
\[ \boxed{\operatorname{Range}=[0,9)}. \]Example: Endpoint Exclusion May Not Exclude the Same Output
Now consider
\[ y=x^2, \qquad -3The output \(4\) occurs at both \(x=-2\) and \(x=2\), so it remains in the range even though one endpoint convention might exclude another point with the same height.
The range is
\[ \boxed{[0,9)}. \]Why Domain and Range Matter
Domain and range summarize the possible inputs and outputs of a relation.
They help us answer questions such as:
- which inputs are permitted;
- which outputs are achievable;
- whether a model is bounded;
- whether a value is impossible;
- whether a graph has gaps or restrictions;
- how transformations or modeling constraints change allowable values.
Domain and Range Describe Extent Before Behavior
Domain and range tell us where a graph exists horizontally and vertically.
They do not by themselves tell us whether the graph is increasing, decreasing, flat, or turning within those sets.
Those questions belong to the next stage of graphical analysis.
Practice 1: Finite Points
Find the domain and range of the relation
\[ \{(-4,2),(-1,5),(2,2),(6,-3)\}. \]Practice 2: Interval with Mixed Endpoints
A continuous graph begins at an open point whose \(x\)-coordinate is \(-5\) and ends at a filled point whose \(x\)-coordinate is \(4\), with no gaps in between. Find the domain.
Practice 3: Unbounded Graph
A graph begins at the filled point \((2,-1)\) and continues indefinitely to the right and upward. State a possible domain and range consistent with that description.
Practice 4: Hole in the Domain
A graph exists for every real \(x\) except \(x=3\). Write the domain in interval notation and set-builder notation.
Practice 5: Horizontal Line
Find the domain and range of
\[ y=-6. \]Practice 6: Vertical Line
Find the domain and range of the relation
\[ x=4. \]Practice 7: Restricted Square-Root Graph
A square-root-shaped graph begins at the filled point \((-2,3)\) and extends indefinitely to the right and upward. Find its domain and range.
Practice 8: Disconnected Domain
A graph exists for
\[ -7\le x<-2 \]and
\[ 1Practice 9: Distinguish Window from Domain
A graphing calculator shows part of the line \(y=3x+1\) in the window \(-10\le x\le10\). What is the domain of the unrestricted line?
Practice 10: Identify the Error
A graph contains the open point \((2,5)\) and the filled point \((2,-1)\). A student says \(2\) is not in the domain because the graph has a hole at \(x=2\). Explain the error.
Practice Answers
Practice 1
Collect the distinct first coordinates for the domain and distinct second coordinates for the range:
\[ \boxed{\operatorname{Domain}=\{-4,-1,2,6\}}, \] \[ \boxed{\operatorname{Range}=\{-3,2,5\}}. \]Practice 2
The left endpoint is excluded and the right endpoint is included:
\[ \boxed{(-5,4]}. \]Practice 3
With the stated endpoint and continued rightward and upward behavior,
\[ \boxed{\operatorname{Domain}=[2,\infty)}, \] \[ \boxed{\operatorname{Range}=[-1,\infty)}. \]Practice 4
All real inputs except \(3\) are allowed:
\[ \boxed{(-\infty,3)\cup(3,\infty)}. \]In set-builder notation,
\[ \boxed{\{x\in\mathbb{R}:x\ne3\}}. \]Practice 5
A horizontal line extends through every real input but has only one output:
\[ \boxed{\operatorname{Domain}=(-\infty,\infty)}, \] \[ \boxed{\operatorname{Range}=\{-6\}}. \]Practice 6
A vertical line uses one input and every real output:
\[ \boxed{\operatorname{Domain}=\{4\}}, \] \[ \boxed{\operatorname{Range}=(-\infty,\infty)}. \]Practice 7
The graph starts at \((-2,3)\) and continues rightward and upward:
\[ \boxed{\operatorname{Domain}=[-2,\infty)}, \] \[ \boxed{\operatorname{Range}=[3,\infty)}. \]Practice 8
Preserve the endpoint conditions on both pieces:
\[ \boxed{[-7,-2)\cup(1,5]}. \]Practice 9
The calculator window displays only part of the line. A nonvertical line extends through every real input:
\[ \boxed{(-\infty,\infty)}. \]Practice 10
The open point removes only \((2,5)\). Because the filled point \((2,-1)\) is included, the graph still contains a point whose first coordinate is \(2\). Therefore,
\[ \boxed{2\text{ is in the domain}}. \]What to Remember
Domain and range summarize the coordinate values represented by a graph.
- domain is the set of represented \(x\)-values;
- range is the set of represented \(y\)-values;
- read domain horizontally and range vertically;
- a value belongs if at least one included graph point uses that coordinate;
- closed endpoints are included and open endpoints are excluded unless that coordinate appears elsewhere;
- arrows indicate continued behavior;
- infinity always uses parentheses;
- disconnected pieces may require unions;
- discrete graphs usually require sets of individual values;
- the graphing window is not automatically the domain or range;
- read the entire graph before excluding an input or output;
- use exact values when known and appropriate estimates when reading from a graph.
The core graphical test is
\[ \boxed{ x\text{ is in the domain}\iff x=a\text{ meets the graph for that }a }, \] \[ \boxed{ y\text{ is in the range}\iff y=b\text{ meets the graph for that }b }. \]Where We Go Next
Domain and range tell us where a graph exists, but they do not yet describe how the graph behaves as we move through those inputs.
The next lesson, Increasing, Decreasing, and Extreme Behavior, develops the language for identifying intervals where outputs rise or fall and for recognizing local and global high and low points directly from graphs.