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Learn to sketch function graphs that satisfy given domain and range intervals, using interval notation and a clear graphing convention.
This lesson teaches you to draw a function graph that meets specific domain and range constraints. You will use interval notation to state the domain and range, and you will follow a consistent graphing convention: mark excluded endpoints with open circles, included endpoints with closed dots, and label axes clearly. The goal is to produce a valid graph, not a single universal answer. You will practice with four self-contained tasks, each solvable from the text.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
The domain is the set of all input values (x-values) for which the function is defined. The range is the set of all output values (y-values) that the function produces. Interval notation uses brackets [ ] for included endpoints and parentheses ( ) for excluded endpoints. For example, [2, 5) means x is at least 2 and less than 5.
When sketching a graph to meet constraints, you must ensure that every x in the domain has a corresponding point on the graph, and that the y-values of all points fall within the range. You may choose any function shape that satisfies these conditions. The graph must be a function: each x in the domain maps to exactly one y.
Use a consistent convention: mark excluded endpoints with open circles and included endpoints with closed dots. Label axes and write interval notation clearly. Different valid function shapes are acceptable as long as the domain and range constraints are met.
There is no single correct graph for a given domain and range. Any function that meets the constraints is valid. For example, a straight line, a curve, or a piecewise function may all work. The rubric focuses on whether the domain and range are correct, endpoints are properly marked, and the graph is a function.
After sketching, verify that for every x in the domain, there is exactly one point on the graph. Then check that the lowest and highest y-values on the graph match the range interval. If the range is [a, b], the graph should reach y = a and y = b, and no point should have y < a or y > b.
Use simple arithmetic to test a few x-values. For instance, if the domain is [1, 4] and the range is [2, 8], pick x = 1, 2, 3, 4 and compute the corresponding y-values from your chosen function. Ensure they lie between 2 and 8, and confirm the graph actually reaches y = 2 and y = 8 at some x in [1, 4]. These numbers are hypothetical examples chosen for easy checking.
Sketch a function with domain [1, 4] and range [2, 8]. Write the domain and range in interval notation.
One valid graph is the line segment from (1, 2) to (4, 8). The domain is [1, 4] and the range is [2, 8]. Endpoints are closed because brackets are used. The graph is a function because each x has exactly one y. Check: at x=2, y=4; at x=3, y=6; both lie in [2, 8].
Sketch a function with domain [0, 3] and range [1, 5]. Write the domain and range in interval notation.
A valid graph is a line segment from (0, 1) to (3, 5). Domain: [0, 3]; Range: [1, 5].
Why: The line segment covers all x from 0 to 3 and produces y-values from 1 to 5. Endpoints are closed, so use closed dots.
Check: Verify that the graph passes through (0,1) and (3,5), and that for x=1.5, y=3, which is within [1,5].
Sketch a function with domain (2, 6) and range [0, 4]. Write the domain and range in interval notation.
One valid graph is a piecewise-linear function made of three connected segments: from (2, 1) down to (3, 0), then up to (5, 4), then down to (6, 1). Use open circles at (2, 1) and (6, 1) because x=2 and x=6 are excluded. The lowest point is (3, 0), so y=0 is attained; the highest point is (5, 4), so y=4 is attained. Domain: (2, 6); Range: [0, 4].
Why: The domain excludes 2 and 6, so use open circles at x=2 and x=6. The range includes 0 and 4, so the graph must reach y=0 and y=4 at some x in the domain. Since x=2 and x=6 are excluded, attain y=0 at the interior x=3 and y=4 at the interior x=5. The three segments form a single function: each x in (2, 6) has exactly one y-value.
Check: Check that the graph does not include x=2 or x=6 (open circles), that some interior x gives y=0 and some interior x gives y=4, and that all y-values lie in [0, 4].
Sketch a function with domain [-1, 2] and range (-3, 1). Write the domain and range in interval notation.
One valid graph is a line segment from (-1, -2.9) to (2, 0.9), with closed dots at both endpoints. Domain: [-1, 2]; Range: (-3, 1).
Why: The domain includes -1 and 2, so use closed dots at x=-1 and x=2. The range excludes -3 and 1, so all y-values must be strictly between -3 and 1. A line from (-1, -2.9) to (2, 0.9) stays strictly inside the range. The values -2.9 and 0.9 are hypothetical examples chosen for easy checking.
Check: Verify that at x=-1, y=-2.9 is greater than -3, and at x=2, y=0.9 is less than 1. Check that no y-value equals -3 or 1, and that the graph is defined for every x in [-1, 2].
Sketch a function with domain [0, 5) and range [2, 6]. Write the domain and range in interval notation.
One valid graph is a curve that starts at (0, 2) with a closed dot, rises to reach y=6 at an interior x such as x=4, then descends to approach (5, 3) with an open circle, staying within y between 2 and 6. Domain: [0, 5); Range: [2, 6].
Why: The domain includes 0 but excludes 5, so use a closed dot at x=0 and an open circle at x=5. The range includes 2 and 6, so the graph must reach y=2 and y=6 at some x in the domain. Since x=5 is excluded, attain y=6 at an interior x such as x=4. The value x=4 is a hypothetical example chosen for easy checking.
Check: Check that the graph includes x=0 and does not include x=5 (open circle), that the minimum y is 2 (attained at x=0) and maximum y is 6 (attained at x=4), and that all y-values lie in [2, 6]. Test an interior value such as x=2 to confirm its y-value is within [2, 6].
Yes, as long as the graph is a function and its domain and range match the given intervals. Linear, quadratic, or piecewise functions are all acceptable.
Use an open circle at the endpoint. For included endpoints, use a closed dot.
The range interval specifies the minimum and maximum y-values. Your graph should attain those values if the endpoints are included; if excluded, the graph should approach them without reaching them.
Yes, writing the domain and range in interval notation is part of the task and helps verify your graph.