The retail price of a certain item at a store is x dollars, where x > 5 If a coupon is used that reduces the retail price by $5, the reduced price r, in dollars, of the item is given by the function r(x) = x - 5 for x > 5 After a 5 percent sales tax is charged by the store on the reduced price of the item, the cost c, in dollars, of the item is given by the function c(r) = 1.05r Which of the following gives the cost c of the item as a function of its retail price x for x > 5 ?

Rationale

c(x) = 1.05(x - 5)

To find the cost of the item as a function of its retail price x, we first need to compute the reduced price r using the function r(x) = x - 5. After applying the 5 percent sales tax, the cost function c becomes c(r) = 1.05r. By substituting r with (x - 5), we arrive at the function c(x) = 1.05(x - 5).

A (c(x) = 1.05x - 5) YOUR ANSWER

This option incorrectly suggests that the cost is simply the retail price multiplied by 1.05 with a direct subtraction of 5. It does not account for the necessary reduction in price due to the coupon before applying the sales tax, leading to an incorrect formula.

B (c(x) = 1.05(x - 5)) CORRECT

This choice accurately reflects the process of first applying the coupon to the retail price, yielding the reduced price r. It then correctly applies the sales tax to that reduced price, providing the accurate cost function for the item.

C (c(x) = (1.05x)(x - 5)) YOUR ANSWER

This option incorrectly implies that the cost is a product of the retail price multiplied by 1.05 and the reduced price. This misinterpretation leads to an incorrect representation of the relationship between the retail price, reduced price, and tax.

D (c(x) = 1.05x + (x - 5)) YOUR ANSWER

This choice suggests adding the retail price after tax to the reduced price, which does not accurately represent the calculation process. The sales tax should be applied to the reduced price, not added to the full retail price.

E (c(x) = (1.05x)/(x - 5)) YOUR ANSWER

This option presents a fraction that incorrectly divides the retail price multiplied by the sales tax by the reduced price. This formulation does not reflect any correct pricing or tax application method.

Conclusion

The cost function c(x) = 1.05(x - 5) correctly describes the relationship between the retail price and the final cost after applying a coupon and sales tax. It ensures that the reduction is applied before the tax calculation, accurately reflecting the pricing process for items sold with discounts. All other options either misrepresent the calculation or fail to follow the correct order of operations.

c(x) = 1.05(x - 5)

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