Combine standard function types arithmetically to create models.
Build a contextual sum or difference of functions
Problem
Revenue and cost are \(R(p)=p(40-p)\) and \(C(p)=100+2p~\text{dollars}\). Build \(P(p)=R(p)-C(p)\), show the entire cost expression grouped before distributing the negative sign, simplify the model, and interpret positive, zero, and negative outputs.
Big Picture
What this problem is really about
Profit is a directed difference, so subtracting the cost function must reverse every term inside its grouping. We’ll form revenue minus the complete cost, distribute the subtraction, simplify the quadratic, and interpret positive, zero, and negative outputs as the three possible revenue-cost relationships.
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