Is the Pattern a Function? Domain, Range, and a Trap
Students are given a relation defined by a table and asked whether it is a function, then to describe its domain and range. The relation pairs each student in a 4-person group with the number of pets they own: Ana→2, Ben→2, Cara→0, Dev→2. A student, Marcus, says "It's not a function because three people all map to 2-the output repeats." Another student, Lena, says "It is a function, and the range is {0, 2} but the domain is {2} because that's the only output used." The teacher wants to resolve both confusions and connect them to the definition of a function.
1. Evaluate Marcus's reasoning against the definition of a function, using the specific pairs.
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Marcus has the definition backwards. A relation is a function when each input (x-value) is assigned exactly one output-repeated outputs are perfectly fine; repeated inputs with different outputs are the problem.
Check each input:
- Ana→2 (one output)
- Ben→2 (one output)
- Cara→0 (one output)
- Dev→2 (one output)
No input is paired with two different outputs, so this is a function. The fact that Ana, Ben, and Dev all map to 2 is allowed-many inputs may share an output (e.g., the constant function f(x)=2 does exactly this). Marcus confused "outputs repeat" (allowed) with "an input repeats with different outputs" (not allowed). A correct non-example would be Ana→2 and Ana→5.
2. Evaluate Lena's domain/range claim and give the correct domain and range.
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Lena has swapped domain and range. The domain is the set of inputs; the range is the set of outputs actually attained.
- Inputs: {Ana, Ben, Cara, Dev} → domain = {Ana, Ben, Cara, Dev}.
- Outputs: {2, 2, 0, 2} → distinct values → range = {0, 2}.
Lena correctly listed {0, 2} but assigned it to the wrong side, then invented a domain of {2} from the most common output. The domain is never a set of outputs. A useful check: the domain answers "what can go in?" and the range answers "what comes out?" Here inputs are students, outputs are pet counts, so the domain must be students and the range must be numbers.