Problem 102 - College Algebra Essentials

Question

The regular price of a pair of jeans is \(x\) dollars. Let \(f(x)=x-5\) and \(g(x)=0.6 x\) a. Describe what functions \(f\) and \(g\) model in terms of the price of the jeans. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the jeans. c. Repeat part (b) for \((g \circ f)(x)\) d. Which composite function models the greater discount on the jeans, \(f \circ g\) or \(g \circ f ?\) Explain.

Step-by-Step Solution

Answer

a. \(f(x)\) represents a flat $5 off and \(g(x)\) represents a 40% discount from the regular price. \n b. \((f \circ g)(x)\) is \(0.6x-5\) which indicates a first 40% discount and then a $5 off.\n c. \((g \circ f)(x)\) is \(0.6(x - 5)\) which indicates a $5 off first and then a 40% discount. \n d. The composite function \((f \circ g)(x)\) models the greater discount on the jeans.

Step 1: Understanding Functions \(f\) and \(g\)

Function \(f(x) = x - 5\) represents a flat $5 discount on the price of the jeans. While function \(g(x) = 0.6x\) represents a 40% (since \(1-0.6 = 0.4\)) discount on the regular price.

Step 2: Finding \((f \circ g)(x)\)

\((f \circ g)(x)\) means applying function \(g\) first, then function \(f\). So, \((f \circ g)(x) = f(g(x)) = f(0.6x) = 0.6x - 5\.

Step 3: Interpreting \((f \circ g)(x)\)

\((f \circ g)(x) = 0.6x - 5\) represents a situation where the jeans are first discounted by 40%, then a flat $5 discount is given.

Step 4: Finding \((g \circ f)(x)\)

\((g \circ f)(x)\) means applying function \(f\) first, then function \(g\). So, \((g \circ f)(x) = g(f(x)) = g(x - 5) = 0.6(x - 5)\.

Step 5: Interpreting \((g \circ f)(x)\)

\((g \circ f)(x) = 0.6(x - 5)\) represents a situation where the jeans first get a flat $5 off, then whatever is left is discounted by 40%.

Step 6: Comparing Discounts

The composite function \((f \circ g)(x)\) gives a bigger discount. This is because it reduces the price by 40% first which is likely a larger amount then it reduces a further $5. In contrast, \((g \circ f)(x)\) removes $5 first, which may result in a smaller amount being reduced when the 40% reduction is done second.