**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**
The daily cost to your donut service to prepare $x$ boxes of organic donuts is

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

(which happens to be a linear function). The fixed cost is $\$100$, the variable cost is $2x,$ and the marginal cost is $2.$

**
Revenue function **
The

**revenue** resulting from one or more business transactions is the total payment received, sometimes called the

*gross proceeds.* If $R(x)$ is the revenue from selling $x$ items at a price of $m$ each, then $R$ is the linear function $R(x) = mx$ and the selling price $m$ can also be called the

**marginal revenue.**
**Example**
Your donut service sells organic donuts for $\$4.50$ per box. Thus, the revenue from the sale of $x$ boxes is

$R(x) = 6.50x$ $\qquad$ \t

The marginal revenue is $m = \$4.50$ per box.

**#[Profit Function][Función de ganancia]#**
The

**profit** is the net proceeds, or what remains of the revenue when costs are subtracted. If the profit depends linearly on the number of items, the slope $m$ is called the

**marginal profit.** Profit, revenue, and cost are related by the following formula.

#[Profit][Ganancia]# \t ${}={}$ #[Revenue − Cost][Ingreso − Costo]#
\\ #[$P$][$G$]# \t #[${}= R - C$][${}= I - C$]#

If the profit is negative, say &minus\$500, we refer to a

**loss** (of \$500 in this case). To

**break even** means to make neither a profit nor a loss. Thus, break even occurs when $P = 0,$ or

#[$R = C \iff P = 0$][$I = C \iff G = 0$]# \t \t

The

**break even point** is the number of items $x$ at which break even occurs.

Continuing with the donut scenario: Given the above cost and revenue functions, the profit function is

#[$P(x){}$][$G(x){}$]# \t #[${}= R(x) - C(x)$][${}= I(x) - C(x)$]#
\\ \t ${}= 4.5x - (2x + 100)$
\\ \t ${}= 2.5x - 100$

For break even, we set $P = 0:$.

$P {}= 2.5x - 100 = 0$${}\implies x = \dfrac{100}{2.5} = 40$ #[boxes][cajas]#

#[So, to avoid a loss, you would need to sell at least 40 boxes of organic donuts.][Entonces, para evitar pérdidas, necesitarías vender 40 cajas de donas orgánicas.]#