Restrict a Function to Meaningful Counts

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A fictional box holds at most 7 whole markers. The input counts how many markers are in it. Which domain fits? Candidate answers: (A) All positive real numbers (B) All real numbers from 0 to 7 (C) Integers 0 through 7 (D) Integers greater than 7.
• Independent check inputs: A fictional box holds at most 8 whole markers. The input counts how many markers are in it. Which domain fits? Candidate answers: (A) All positive real numbers (B) All real numbers from 0 to 8 (C) Integers 0 through 8 (D) Integers greater than 8.

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. A counting context supplies input restrictions beyond a formula. Whole-item counts are integers, and a capacity constraint creates a largest allowed input while an empty collection may allow zero.

Here is the method on the supplied teaching case. You cannot have a fractional marker in this counting model, and an empty box is possible. The capacity makes 7 the largest allowed input.

Do not include fractions just because an algebraic formula can evaluate them. State whether the context is discrete or continuous and whether its endpoint is included; physical capacity is part of the input definition.

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: A fictional box holds at most 14 whole markers. The input counts how many markers are in it. Which domain fits? Candidate answers: (A) All positive real numbers (B) All real numbers from 0 to 14 (C) Integers 0 through 14 (D) Integers greater than 14. Show the mathematical reason, not only the selected answer. Expected result: Integers 0 through 14. You cannot have a fractional marker in this counting model, and an empty box is possible. The capacity makes 14 the largest allowed input.

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