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revenue function — Definition, Formula & Examples

A revenue function gives the total income earned from selling a certain number of units of a product. It is typically written as R(x)R(x), where xx is the quantity sold.

The revenue function R(x)R(x) maps the quantity of goods sold to the total income received, defined as the product of the price per unit p(x)p(x) and the number of units sold xx. When price is constant, R(x)=p⋅xR(x) = p \cdot x; when price depends on demand, R(x)=p(x)⋅xR(x) = p(x) \cdot x, where p(x)p(x) is the demand (or price-demand) function.

Key Formula

R(x)=p(x)⋅xR(x) = p(x) \cdot x
Where:
  • R(x)R(x) = Total revenue from selling x units
  • p(x)p(x) = Price per unit, which may be constant or a function of x
  • xx = Number of units sold

How It Works

To build a revenue function, you need to know the selling price and the quantity sold. If every unit sells at a fixed price, just multiply price by quantity. In more realistic models, the price depends on how many units are sold—described by a demand function p(x)p(x)—so you multiply that expression by xx. Revenue is then combined with a cost function to find profit: P(x)=R(x)−C(x)P(x) = R(x) - C(x).

Worked Example

Problem: A company sells headphones. Market research shows the price-demand function is p(x)=120−2xp(x) = 120 - 2x dollars, where xx is the number of units sold (in hundreds). Find the revenue function and calculate the revenue when 25 hundred units are sold.
Write the revenue function: Multiply the price-demand function by the quantity xx.
R(x)=p(x)⋅x=(120−2x) x=120x−2x2R(x) = p(x) \cdot x = (120 - 2x)\,x = 120x - 2x^2
Substitute x = 25: Plug in x=25x = 25 (representing 2,500 units).
R(25)=120(25)−2(25)2=3000−1250=1750R(25) = 120(25) - 2(25)^2 = 3000 - 1250 = 1750
Answer: The revenue function is R(x)=120x−2x2R(x) = 120x - 2x^2, and selling 2,500 units generates $1,750 (in hundreds of dollars, i.e., $175,000).

Why It Matters

Revenue functions appear throughout business calculus and microeconomics. You use them to find the quantity that maximizes revenue (by setting R′(x)=0R'(x) = 0), and they combine directly with cost functions to determine break-even points and maximum profit—core tasks in managerial decision-making.

Common Mistakes

Mistake: Confusing revenue with profit by ignoring costs.
Correction: Revenue is total income before expenses. Profit equals revenue minus cost: P(x)=R(x)−C(x)P(x) = R(x) - C(x). Always subtract the cost function to get profit.

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