Check Whether Every Input Has One Output

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Among the supplied sets using inputs 2, 3 and 4, which defines y as a function of x? Candidate answers: (A) (2,1), (2,2), (3,3) (B) (2,1), (3,1), (4,2) (C) (3,0), (3,1), (4,2) (D) (4,2), (4,3), (2,1).
• Independent check inputs: Among the supplied sets using inputs 3, 4 and 5, which defines y as a function of x? Candidate answers: (A) (3,1), (3,2), (4,3) (B) (3,1), (4,1), (5,2) (C) (4,0), (4,1), (5,2) (D) (5,2), (5,3), (3,1).

Learn the idea

A function assigns one output to each allowed input. A table can reveal nonlinear structure or interval-average change, while the domain records which inputs make sense mathematically and in the supplied context. Repeated outputs are permitted in a function, but a repeated input cannot be paired with two different outputs. Inspecting input assignments tests the definition directly.

Here is the method on the supplied teaching case. The correct set assigns each of the three distinct inputs 2, 3 and 4 exactly one output. Repeated output 1 is allowed; one input with two outputs is not.

The rule concerns one output per input, not one input per output. Group pairs by their first coordinate before judging the relation; two different inputs may legitimately share an output.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Among the supplied sets using inputs 9, 10 and 11, which defines y as a function of x? Candidate answers: (A) (10,0), (10,1), (11,2) (B) (11,2), (11,3), (9,1) (C) (9,1), (10,1), (11,2) (D) (9,1), (9,2), (10,3). Show the mathematical reason, not only the selected answer. Expected result: (9,1), (10,1), (11,2). The correct set assigns each of the three distinct inputs 9, 10 and 11 exactly one output. Repeated output 1 is allowed; one input with two outputs is not.

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