Problem

\(C(n)\) gives total cost in dollars for \(n\) items on a domain where \(C\) is strictly increasing. Describe the forward and inverse mappings with units, and explain why the inverse item count is unique.

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Interpret an inverse by reversing the quantities while keeping each quantity’s unit attached to it. An original output becomes the inverse input, and the corresponding original input becomes the inverse output. Strict increase matters because it prevents two different original inputs from sharing one output, so the reversed mapping recovers a unique quantity.

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Forward: C maps item count, in items, to total cost, in dollars. Inverse: C⁻¹ maps cost c, in dollars, to the unique item count, in items, whose cost is c. Uniqueness follows because C is strictly increasing.

Forward: C maps item count, in items, to total cost, in dollars. Inverse: C⁻¹ maps an item count, in items, to total cost, in dollars. Uniqueness follows because C is strictly increasing.

Forward: C maps item count, in items, to total cost, in dollars. Inverse: C⁻¹ maps cost c, in dollars, to several possible item counts, in items. Strict increase does not provide uniqueness.

Forward: C maps item count, in dollars, to total cost, in items. Inverse: C⁻¹ maps cost c, in items, to the unique item count, in dollars, whose cost is c. Uniqueness follows because C is strictly increasing.

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\(C(n)\) gives total cost in dollars for \(n\) items on a domain where \(C\) is strictly increasing. Describe the forward and inverse mappings with units, and explain why the inverse item count is unique.

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