Let g(a) represent the number of gallons of sealant needed...

Let $g(a)$ represent the number of gallons of sealant needed to seal a bamboo floor with area $a$. Let $c(s)$ represent the cost of $s$ gallons of sealant. Which composition makes sense: $(c \circ g)(a)$ or $(g \circ c)(s) ?$ What does it represent?

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Key Concepts

Function Composition

This concept involves linking two functions where the output of one function becomes the input of another. Here, the function g calculates the number of gallons of sealant required for a given floor area, and the function c then determines the cost based on the gallons used. Composing c after g, as in (c ? g)(a), enables you to determine the total cost directly from the floor area.

Domain and Range

Understanding the domains and ranges of functions is essential when composing them. The function g accepts an area as input and outputs the number of gallons needed, which falls within the domain of c (since c requires a number of gallons as input). This alignment makes the composition (c ? g)(a) valid, whereas (g ? c)(s) would not be meaningful because c’s output (cost) does not match the expected input for g.

Real-World Modeling

This problem is an example of using mathematical functions to model a real-world process. It shows how multiple steps—determining the amount of sealant required based on area and then calculating the cost based on that amount—can be combined into one overall function. The composition (c ? g)(a) serves as a tool to predict total cost directly from the area of the bamboo floor.

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