**Cost function**
A

**cost function** specifies the cost $C$ as a function of the number of items $x.$ Thus, $C(x)$ is the cost of $x$ items, and has the form

Cost = Variable Cost + Fixed Cost

where the variable cost is a function of $x$ and the fixed cost is a constant. A cost function of the form

$C(x) = mx + b$

is called a

**linear cost function;** the variable cost is $mx$ and the fixed cost is $b.$ The slope $m$ in a linear cost function is the

**marginal cost,** and measures the incremental cost per item.

**
Example **
Your long-distance phone service charged you a $\$100$ initiation fee and charges an additional $\$2$ per call. Then the cost of making $x$ long-distance calls is

$C(x) = 2x + 100\qquad$ \t

(which happens to be a linear function) is a sum of two parts: a constant, or

**fixed cost,** $\$100$, which is the same regardless of the number $x$ of long-distance calls, or "items" being purchased, and a

**variable cost,** $2x,$ which does depend on the number of items purchased:

Cost = Variable Cost + Fixed Cost

The quantity 2 by itself is the incremental cost per call; we call 2 the

**marginal cost.** The fixed cost 100 is the $C$

*-intercept* of the linear cost function.