Domain and Range From a Graph

You have a graph in front of you, and the problem says 'find the domain and range from a graph.' Your pencil is hovering over the x-axis. The domain is the set of all x-values the graph covers, and the range is the set of all y-values the graph covers. The method is mechanical: project the graph onto the x-axis for domain, then project it onto the y-axis for range. That is the whole idea, and every graph type uses the same two steps.

Imagine shining a flashlight horizontally from the left of the graph. The shadow the graph casts onto the x-axis is its domain. If the graph touches an x-value, that x-value is in the domain. If there is a gap, a hole, or an asymptote at an x-value, that value is excluded. The domain of a line that goes forever left and right is all real numbers, written as (-∞, ∞) in interval notation. The domain of a parabola that opens upward or downward is also all real numbers, because the parabola extends infinitely left and right. Every straight line and every polynomial with an odd degree share this property: domain all reals.

The failure case is when you see a denominator in the equation or a square root. The graph itself will show you a vertical asymptote, a dashed line the curve approaches but never touches, or a gap where the function stops. A rational function like f(x) = 1/x has a vertical asymptote at x = 0. The graph goes to +∞ on the right of zero and -∞ on the left, but it never touches x = 0. The domain is (-∞, 0) ∪ (0, ∞). A square root function like f(x) = √(x - 2) has a graph that starts at x = 2 and goes right forever. The domain is [2, ∞). OpenStax Precalculus 2e, Section 3.2 covers this as the first step in domain analysis, and the Common Core State Standard HSF-IF.A.1 defines a function by its domain exactly this way.

Now shine the flashlight vertically from above the graph. The shadow it casts onto the y-axis is the range. Every y-value the graph reaches is in the range. A line with a slope that is not zero has a range of all real numbers, (-∞, ∞).A parabola that opens upward has a range starting at its vertex y-coordinate and going to ∞.A parabola that opens downward has a range (-∞, vertex y].

The mistake students make most often here is assuming the range is always all real numbers. That claim is true for linear functions with non-zero slope and for odd-degree polynomials, but it is false for quadratics, absolute value functions, exponentials, and sine/cosine. For a quadratic, the vertex y-coordinate is the boundary. For an absolute value function, the vertex y-coordinate is the minimum (or maximum). For an exponential function with base > 1 and no vertical shift, the range is (0, ∞). The graph never goes below zero. OpenStax Precalculus 2e, Section 3.2 gives this as the standard method, and Section 3.1 defines function notation that makes range explicit.

Below are six graph types you will see on homework and tests. Each one shows the domain projection onto the x-axis and the range projection onto the y-axis. The annotations call out the specific x and y values that define the boundaries.

Linear Function (Non-Zero Slope)

Graph: A straight line sloping upward left to right, extending indefinitely in both directions. The line has no breaks, no holes, and no asymptotes. Domain: (-∞, ∞). Range: (-∞, ∞). Both projections cover the entire axis.

Quadratic Function (Parabola Opening Upward)

Graph: A U-shaped curve with vertex at (2, -3). The curve goes left and right forever, so domain is (-∞, ∞). The lowest y-value is -3 at the vertex, and the curve goes up forever. Range: [-3, ∞). The bracket at -3 indicates the vertex is included.

Square Root Function

Graph: f(x) = √(x - 1). The curve starts at (1, 0) and goes right and upward. The domain is [1, ∞) because the radicand x - 1 must be ≥ 0. The range is [0, ∞) because the principal square root is never negative.

Rational Function With a Vertical Asymptote

Graph: f(x) = 1/(x - 2). The curve has a vertical asymptote at x = 2 (dashed line). The left branch approaches -∞ from the left and +∞ from the right, but never touches x = 2. Domain: (-∞, 2) ∪ (2, ∞). The range is all real numbers except y = 0, because the curve approaches 0 from above and below but never reaches it. Range: (-∞, 0) ∪ (0, ∞).

Absolute Value Function

Graph: V-shaped curve with vertex at (-1, 2). The left arm has slope -1, the right arm has slope +1. Domain: (-∞, ∞). The lowest y-value is 2, from the vertex upward. Range: [2, ∞).

Piecewise Function With an Open Circle

Graph: Two line segments. The left segment goes from (-3, 1) to (0, 4) with a closed circle at (0, 4). The right segment goes from (0, 2) with an open circle to (3, 5) with a closed circle. Domain: [-3, 0] ∪ (0, 3]. The open circle at x = 0 on the right segment excludes x = 0 from the domain for that piece, but the closed circle at (0, 4) on the left segment includes x = 0 for that piece. The overall domain includes x = 0 because the left segment covers it. Range: The y-values on the left segment go from 1 to 4, inclusive. The right segment goes from just above 2 to 5, including 5 but not 2. Range: [1, 4] ∪ (2, 5].

A closed dot (filled circle) at an endpoint means that x-value (or y-value) is included in the domain (or range). An open dot (empty circle) means that exact value is excluded. An arrow on the end of a curve means the graph continues indefinitely in that direction. If the graph has an arrow pointing right, the domain extends to ∞. An arrow pointing up means the range extends to ∞. An arrow pointing down means the range extends to -∞. These symbols are how the graph tells you about open and closed intervals without writing a single bracket.

The failure case: a graph that ends with an open circle at x = 3 and an arrow pointing right. The open circle at 3 means 3 is excluded, but the arrow means values just to the right of 3 are included. That domain is (3, ∞). A graph that ends with a closed circle at x = -2 and an arrow pointing left has domain (-∞, -2]. The closed circle at -2 includes -2.

A vertical asymptote is a dashed vertical line that the graph never crosses. The x-value at that line is excluded from the domain. A rational function like f(x) = 1/(x - 2) has a vertical asymptote at x = 2. The domain is (-∞, 2) ∪ (2, ∞). A hole is a single point missing from the graph, shown as an open circle on a point that would otherwise be filled. A hole occurs when a factor cancels in the numerator and denominator. The x-value of the hole is excluded from the domain, but there is no vertical asymptote there because the factor cancels. The range also excludes the y-value of the hole if the function is not defined there.

A horizontal asymptote is a dashed horizontal line that the graph approaches as x goes to ±∞. The y-value of the horizontal asymptote may or may not be excluded from the range. For f(x) = 1/(x - 2), the horizontal asymptote is y = 0, and the range excludes 0 because the graph never reaches it. For f(x) = x/(x² + 1), the horizontal asymptote is also y = 0, but the graph crosses it at x = 0. The range includes 0. The rule is: the range excludes the horizontal asymptote value only if the function never attains that value. The research from OpenStax Precalculus 2e, Section 5.6 covers this distinction.

Work through these three practice graphs. Each one is a different type. Write the domain and range in interval notation, then check your answers below.

Practice Graph 1: Line With an Open Circle

A horizontal line at y = 5 from x = -2 (closed circle) to x = 4 (open circle). The line extends left from -2 with an arrow. Domain: (-∞, 4). The open circle at 4 excludes 4. Range: {5}. Only one y-value appears.

Practice Graph 2: Parabola Opening Downward

Vertex at (1, 4). The parabola opens downward. The graph goes left and right forever. Domain: (-∞, ∞). Range: (-∞, 4]. The vertex y-coordinate is the maximum, and the bracket includes 4.

Practice Graph 3: Piecewise With Three Pieces

Piece 1: f(x) = -x for x in [-3, 0), closed circle at (-3, 3), open circle at (0, 0). Piece 2: f(x) = 2 for x in [0, 2], closed circles at both ends. Piece 3: f(x) = x for x in (2, 5], open circle at (2, 2), closed circle at (5, 5). Domain: [-3, 5]. Piece 1 covers [-3, 0), piece 2 covers [0, 2], piece 3 covers (2, 5]. The union is [-3, 5]. Range: Piece 1 gives y-values from 0 (excluded) up to 3 (included): (0, 3]. Piece 2 gives y = 2. Piece 3 gives y-values from just above 2 to 5: (2, 5]. Union: (0, 3] ∪ {2} ∪ (2, 5] = (0, 5]. The 2 is inside (0, 5), so the range is (0, 5].

Below is a quick worksheet-style list of five function types. For each one, write the domain and range from the graph description. No graph is drawn; the description is all you need. This is the same format as many domain and range graph worksheet exercises in textbooks.

1. A straight line with a closed circle at (-1, 2) and an arrow pointing right. Slope is 0 (horizontal).
2. A cube root function. The curve passes through (-8, -2), (0, 0), and (8, 2). No breaks.
3. A rational function with vertical asymptotes at x = -1 and x = 3. Horizontal asymptote at y = 0. The graph crosses the horizontal asymptote at x = 0.
4. An absolute value function with vertex at (2, -3). Opens upward.
5. A constant function: a horizontal line at y = -4 from x = -5 (closed circle) to x = 2 (closed circle).

Answers: 1. Domain: [-1, ∞), Range: {2}. 2. Domain: (-∞, ∞), Range: (-∞, ∞). 3. Domain: (-∞, -1) ∪ (-1, 3) ∪ (3, ∞). Range: (-∞, ∞) because the graph crosses y = 0 at x = 0, and goes to ±∞ on each side of the asymptotes. 4. Domain: (-∞, ∞), Range: [-3, ∞). 5. Domain: [-5, 2], Range: {-4}.

Graph Type	Domain	Range	Key Feature to Check
Linear (non-zero slope)	(-∞, ∞)	(-∞, ∞)	Check for open circles at endpoints
Quadratic (parabola)	(-∞, ∞)	[vertex y, ∞) or (-∞, vertex y]	Find vertex y-coordinate
Square root	[starting x, ∞)	[starting y, ∞)	Radicand ≥ 0
Rational with vertical asymptote	(-∞, a) ∪ (a, ∞)	Depends on asymptote crossing	Vertical asymptote at x=a
Absolute value	(-∞, ∞)	[vertex y, ∞) or (-∞, vertex y]	Vertex y is the boundary
Constant (horizontal line)	(-∞, ∞) or restricted	{c} (single value)	Only one y-value

The most frequent error is assuming the range is always all real numbers. This claim fails for every even-degree polynomial, every absolute value function, every exponential, and every sine/cosine function. Test one y-value outside your guess: if the graph does not reach it, your guess is wrong.

The second failure is ignoring open circles at endpoints. A graph with an open circle at (2, 4) means x = 2 is not in the domain, and y = 4 is not in the range for that piece. Write parentheses, not brackets, around that endpoint in interval notation.

The third failure is forgetting to use the union symbol when the domain or range has a gap. Writing (-∞, 2) (2, ∞) without the ∪ is incorrect. A single interval with a gap is not a valid interval. The correct notation is (-∞, 2) ∪ (2, ∞).

The fourth failure is assuming the graph shows the entire function. A graph drawn on a finite window (x from -10 to 10, y from -10 to 10) may cut off the function. If you see arrows on the ends, the function continues. If you see no arrows and a curve stops at a point, check whether that point is the actual end of the function (closed or open circle) or just the edge of the window. The Common Core State Standard HSF-IF.B.5 expects you to relate the graph to its domain and range, which means you must reason beyond the window.

A line that is not horizontal has domain and range both all real numbers. A horizontal line has domain all real numbers (if it extends in both directions) and range a single y-value. A vertical line is not a function because it fails the vertical line test, it would have many y-values for one x, so it is not covered by the definition in Common Core HSF-IF.A.1. The domain and range of a line are the simplest case, but students often overgeneralize from lines to all functions. The claim that 'range is always all real numbers' survives only until you encounter a quadratic.

OpenStax College Algebra 2e, Section 3.2 covers domain and range of linear, quadratic, and radical functions explicitly. OpenStax Precalculus 2e, Section 3.1 defines function notation and the requirement that each input maps to exactly one output, which is why vertical lines are excluded.

An open circle on a graph means that specific x-value (or y-value) is not included in the domain (or range). A closed circle means it is included. The key is to look at the endpoint of each piece or curve. If the graph stops at a point with a closed circle, that point's x and y are in the domain and range. If it stops with an open circle, they are not.

On a piecewise function, you may have a closed circle at one x-value on one piece and an open circle at the same x-value on another piece. The domain includes that x-value because the closed circle exists. The range includes the y-value from the closed circle but not the y-value from the open circle.