The members of a health spa pay annual membership dues of $300 plus $1 for each visit to the spa.

Describe the relationship between the number of visits by a member during the year and the total cost of membership.

c002.p0250.q003

Question

The members of a health spa pay annual membership dues of $300 plus $1 for each visit to the spa.

Describe the relationship between the number of visits by a member during the year and the total cost of membership.

c002.p0250.q003

BlackTom AI · Accepted Answer

A function assigns exactly one output to each input; here the number of visits (input) determines a unique total cost (output). The relationship is linear and can be written as C(v) = 300 + 1·v, where v = number of visits and C(v) = total annual cost in dollars.

1. Relevant knowledge points and concepts
- Definition of a function: each input has exactly one output.
- Independent vs dependent variable: v (visits) is independent, C (cost) is dependent.
- Linear function form: C = m·v + b, where m is slope (cost per visit) and b is y-intercept (fixed annual dues).
- Proportional vs non-proportional linear: proportional means b = 0; here b ≠ 0.

2. Step-by-step reasoning
- The spa charges a fixed annual fee of $300 regardless of visits → constant term b = 300.
- Each visit adds $1 to the total → slope m = 1.
- Combine these to form the rule: C(v) = 300 + 1·v.
- For every possible v (0, 1, 2, ...), this rule gives exactly one C value, fulfilling the function requirement.

3. Intermediate calculations (examples)
- v = 0 → C(0) = 300 + 1·0 = $300.
- v = 10 → C(10) = 300 + 1·10 = $310.
- v = 100 → C(100) = 300 + 1·100 = $400.
Each input yields one definite output.

4. Why the correct answer is correct
Because the relationship maps each number of visits to a single, well-defined total cost via the linear rule C(v) = 300 + v, it meets the definition of a function. The fixed fee and per-visit charge produce a deterministic mapping.

5. Why other interpretations are incorrect
- Not proportional: proportional would require no fixed fee (C = k·v); here there is a $300 fixed fee, so it is non-proportional linear.
- Not inverse variation: inverse would be of form C·v = constant, which does not match an additive fee structure.
- Not “not a function”: there is no input that gives multiple outputs, so it is indeed a function.

6. Conclusion
This relationship is functional; the correct answer is Functional.

The members of a health spa pay annual membership dues of $300 plus $1 for each visit to the spa. Describe the relationship between the number of visits by a member during the year and the total cost of membership. c002.p0250.q003单项选择题

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