Problem

For the model \(C(n)~=~8n\), where \(n\) is the number of tickets sold in a \(100\text{-}\text{seat}\) theater, compare the mathematical domain and the contextual domain.

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Mathematical and contextual domains answer different questions. We’ll first inspect the formula alone for algebraic restrictions, then restore the meaning of the input and apply the real-world rules for sign, whole-number counts, and capacity. This two-pass approach keeps physical limitations from being mistaken for algebraic ones while still narrowing the model to inputs the theater can actually use.

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The mathematical domain and the contextual domain are both all real numbers, since C(n)=8n is a linear rule and linear functions accept every real input.

The mathematical domain is all real numbers because the rule C(n)=8n has no algebraic restriction, but the contextual domain is the integers from 0 to 100 because ticket sales must be whole numbers and cannot exceed the 100 seats.

The mathematical domain is the integers from 0 to 100 because the theater has 100 seats, but the contextual domain is all real numbers because the expression 8n can be evaluated for any input.

The mathematical domain and the contextual domain are both the integers from 0 to 100, because n stands for tickets sold and the model should only use values that make sense in the theater.

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For the model \(C(n)~=~8n\), where \(n\) is the number of tickets sold in a \(100\text{-}\text{seat}\) theater, compare the mathematical domain and the contextual domain.

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