Problem 63 - College Algebra and Calculus: An Applied Approach

Question

The suggested retail price of a new hybrid car is \(p\) dollars. The dealership advertises a factory rebate of $$\$ 2000$$ and a \(10 \%\) discount. (a) Write a function \(R\) in terms of \(p\) giving the cost of the hybrid car after receiving the rebate from the factory. (b) Write a function \(S\) in terms of \(p\) giving the cost of the hybrid car after receiving the dealership discount. (c) Form the composite functions \((R \circ S)(p)\) and \((S \circ R)(p)\) and interpret each. (d) Find \((R \circ S)(20,500)\) and \((S \circ R)(20,500)\). Which yields the lower cost for the hybrid car? Explain.

Step-by-Step Solution

Answer

The functions \(R(p)\) and \(S(p)\) are \(R(p) = p - 2000\) and \(S(p) = 0.9p\) respectively. The composite functions are \(R(S(p)) = 0.9p - 2000\) and \(S(R(p)) = 0.9p - 1800\). The cost of the hybrid car is lower when the dealership discount is applied first, at $16,450.

Step 1: Writing function R

We need to write a function \(R(p)\), which gives the cost of the car after receiving the factory rebate of $2000. The function \(R(p)\) is simply the original price \(p\) minus the rebate, which gives \(R(p) = p - 2000\).

Step 2: Writing function S

Next we write a function \(S(p)\), which gives the cost of the car after the discount of 10% by the dealership. The function \(S(p)\) is the original price \(p\) minus 10% of the original price, which gives \(S(p) = p - 0.1p = 0.9p\).

Step 3: Forming Composite Functions

We form the composite functions \(R(S(p))\) and \(S(R(p))\). \(R(S(p))\) represents the cost of the car when the dealership discount is applied first then the factory rebate, which gives \(R(S(p)) = R(0.9p) = 0.9p - 2000\). \(S(R(p))\) represents the cost of the car when the factory rebate is applied first then the dealership discount, which gives \(S(R(p)) = S(p - 2000) = 0.9(p - 2000) = 0.9p - 1800\).

Step 4: Computing the Composite Functions

To find \(R(S(20500))\) and \(S(R(20500))\), we can substitute \(p = 20500\) into the composite functions. \(R(S(20500)) = 0.9(20500) - 2000 = 16450\), and \(S(R(20500)) = 0.9(20500 - 2000) = 0.9(18500) = 16650\). So, applying the dealership discount first then the factory rebate yields a lower cost for the hybrid car.