21. The retail price of an item is x dollars, where x > 5. If a coupon reduces the price by 5, the reduced price r, in dollars, is r(x) = x - 5. After a 5 percent sales tax is charged on the reduced price, the cost c, in dollars, is 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?
Answer: B
c(x) = 1.05(x - 5)
The cost c of the item as a function of its retail price x is accurately represented by the equation c(x) = 1.05(x - 5). This equation correctly reflects the application of the coupon discount followed by the sales tax calculation.
A) c(x) = 1.05x - 5
This option is incorrect because it suggests that the retail price x is directly multiplied by the sales tax rate of 1.05 and then subtracts 5, which does not account for the proper sequence of applying the coupon before calculating the sales tax.
B) c(x) = 1.05(x - 5)
This option is correct as it first subtracts the coupon discount of 5 from the retail price x, resulting in the reduced price r = x - 5. The sales tax is then applied to this reduced price, correctly yielding the cost function c(x) = 1.05(x - 5).
C) c(x) = (1.05x)(x - 5)
This choice is incorrect because it implies that the sales tax is applied to the original price x before any discount is applied. This misrepresentation does not follow the correct order of operations involving the coupon discount followed by the sales tax.
D) c(x) = 1.05x + (x - 5)
This option is incorrect because it adds the full retail price multiplied by the sales tax to the reduced price instead of applying the sales tax to the reduced price alone. This does not reflect the correct final cost.
E) c(x) = (1.05x)/(x - 5)
This option is incorrect as it presents a division of the sales tax adjusted retail price by the reduced price, which does not correspond to the calculation of the final cost after applying the discount and tax.
Conclusion
The function c(x) = 1.05(x - 5) effectively captures the necessary steps of first applying the coupon discount and then calculating the sales tax on the reduced price. Other options either misapply the sequence of operations or fail to correctly reflect the relationship between the retail price and the final cost after tax, confirming that option B is the only accurate representation of the cost function.