• Add, subtract, multiply, and divide functions
• Plug one function into another and evaluate it
• Find the domain of a composite function

Coastal Bites is a food truck that has become a local favorite. Food truck success depends on multiple factors working together – location affects foot traffic, foot traffic affects sales volume, and sales volume affects daily profit. Understanding function composition helps food truck owners optimize their operations by seeing how changes in one area create chain reactions throughout their entire business.

In the food truck industry, business factors are interconnected. For example:

• Location choice affects customer foot traffic
• Foot traffic affects daily sales volume
• Sales volume affects ingredient costs and profit margins

These relationships form function compositions: Daily profit depends on sales volume, which depends on foot traffic, which depends on location choice. Today we’ll model these real business relationships that food truck entrepreneurs navigate every day.

Coastal Bites tracks two main revenue streams during lunch hours:

Number of walk-up customer orders: [latex]W(t) = 25t[/latex] (customers per hour, where t = hours since 11 AM)

Number of office delivery orders: [latex]D(t) = 10t + 10[/latex] (orders per hour, where t = hours since 11 AM)

[latex](f\circ g)=f(g(x))[/latex] is [latex]f[/latex] composed of [latex]g[/latex] of [latex]x[/latex]

Now let’s model how Coastal Bites’ order volume creates a chain reaction through their profit calculations.

Consider a profit function based on the average profit of the items Coastal Bites sells: [latex]P(n)=8n-150[/latex] where [latex]n[/latex] represents the total number of orders.

In calculus, you’ll learn that the derivative of a composite function requires the chain rule. Recognizing when a function is composed of multiple functions is essential for applying this rule correctly. For example, if you wanted to find how quickly revenue changes with respect to location rating, you’d need to identify that revenue depends on foot traffic, which depends on location rating – this creates a “chain” of dependencies that the chain rule handles.

The domain of the composed function, [latex]f\circ g{/latex] is based on the domain of [latex]g(x)[\latex] and which outputs of [latex]g(x)[/latex] fit within the domain of [latex]f(x)[/latex] because [latex]g(x)[/latex] becomes the domain of [latex]f(x)[/latex].