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profit function — Definition, Formula & Examples

A profit function gives the profit earned from producing and selling xx units by subtracting the total cost from the total revenue. It tells you exactly how much money you gain (or lose) at any production level.

Given a revenue function R(x)R(x) and a cost function C(x)C(x), the profit function is defined as P(x)=R(x)−C(x)P(x) = R(x) - C(x), where xx represents the quantity of goods produced and sold. Profit is positive when revenue exceeds cost and negative (a loss) when cost exceeds revenue.

Key Formula

P(x)=R(x)−C(x)P(x) = R(x) - C(x)
Where:
  • P(x)P(x) = Profit from producing and selling x units
  • R(x)R(x) = Total revenue from selling x units
  • C(x)C(x) = Total cost of producing x units
  • xx = Number of units produced and sold

How It Works

To build a profit function, you need two ingredients: a revenue function and a cost function, both written in terms of the same variable xx. Subtract the cost function from the revenue function and simplify. Once you have P(x)P(x), you can set P(x)=0P(x) = 0 to find the break-even points, or use calculus to find the production level that maximizes profit by solving P′(x)=0P'(x) = 0.

Worked Example

Problem: A company sells widgets at $50 each. The cost function is C(x)=2000+20xC(x) = 2000 + 20x, where xx is the number of widgets. Find the profit function and determine the break-even quantity.
Write the revenue function: Revenue equals price times quantity.
R(x)=50xR(x) = 50x
Subtract cost from revenue: Apply the profit formula and simplify.
P(x)=50x−(2000+20x)=30x−2000P(x) = 50x - (2000 + 20x) = 30x - 2000
Find the break-even point: Set the profit function equal to zero and solve for xx.
30x−2000=0  ⟹  x=200030≈66.730x - 2000 = 0 \implies x = \frac{2000}{30} \approx 66.7
Answer: The profit function is P(x)=30x−2000P(x) = 30x - 2000. The company must sell at least 67 widgets to turn a profit.

Why It Matters

In business and economics courses, the profit function is the starting point for optimization problems — finding the production level that maximizes profit. It also appears in managerial accounting and entrepreneurship whenever you need to model whether a product line is financially viable.

Common Mistakes

Mistake: Subtracting revenue from cost instead of cost from revenue, writing P(x)=C(x)−R(x)P(x) = C(x) - R(x).
Correction: Profit is revenue minus cost: P(x)=R(x)−C(x)P(x) = R(x) - C(x). Reversing the order flips the sign, turning profits into losses and vice versa.

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