Lecture 1 · Barnett ch. 1 · about 40 min

Functions and Their Graphs

What a function is, how to read its domain and range, how piecewise rules describe real pricing, and how demand and supply are just functions with an economic name.

By the end you can

Decide whether a rule or a graph defines a function, using the vertical line test
Find the domain and range of a function, including ones with an economic restriction
Evaluate and interpret piecewise-defined functions such as tiered pricing
Build and read demand, supply, cost and revenue functions, and find their inverses

What a function is

A function is a rule that assigns to each input exactly one output. In business, this is the most natural idea in the world: total cost depends on how many units you make, and for a given output level there is exactly one total cost, not two. Write $C = f(x)$ : cost is a function of quantity $x$ .

Function notationFormula sheet →

y = f(x)

$x$: input (independent variable)
$y$: f(x) = output (dependent variable)

Use it when you need to name a relationship before working with it, or evaluate it at a specific input.

The single-output requirement is what makes a rule a function. “Price of a house” is not a function of “number of bedrooms alone”, because two 3-bedroom houses can have different prices. But “cost of producing $x$ phones this month, given the factory’s cost structure” is a function of $x$ : plug in a quantity, get exactly one cost back.

Domain and range

The domain is the set of inputs a function is allowed to take. The range is the set of outputs it actually produces. Two things restrict a domain in business problems: the algebra (you cannot divide by zero or take the square root of a negative number) and the economics (you cannot produce $-5$ units).

Building demand, supply, cost and revenue functions

Economics gives functions their names. A demand function $p = D(x)$ gives the price at which $x$ units will sell; a supply function $p = S(x)$ gives the price at which producers will supply $x$ units. Total revenue is price times quantity, and total cost typically has a fixed part plus a variable part.

Cost, revenue and profitFormula sheet →

C(x) = \text{fixed cost} + (\text{cost per unit})\,x \qquad R(x) = p\,x \qquad P(x) = R(x) - C(x)

$C(x)$: total cost
$R(x)$: total revenue
$P(x)$: total profit
$p$: price per unit

Use it when you are given a fixed cost, a per-unit cost, and a price, and need any of the three functions.

Piecewise-defined functions

Many real prices are piecewise: a different formula applies on different parts of the domain. Tiered electricity pricing, income tax brackets, and bulk-discount pricing are all piecewise functions.

Composition of functions

If output depends on labour, and labour depends on the wage rate, then output depends on the wage rate through labour. This chaining is composition: $(f \circ g)(x) = f(g(x))$ , read ” $f$ of $g$ of $x$ ”: first apply $g$ , then feed the result into $f$ .

CompositionFormula sheet →

(f \circ g)(x) = f\big(g(x)\big)

$g$: the inner function, applied first
$f$: the outer function, applied to g's output

Use it when one quantity is defined in terms of a second, which is itself defined in terms of a third, such as cost as a function of labour as a function of the wage.

Inverse functions

An inverse function undoes another function: if $y = f(x)$ , then $x = f^{-1}(y)$ . Economically, if a demand function gives price from quantity, its inverse gives quantity from price, which is exactly the form used to draw a demand curve with price on the vertical axis.

Inverse functionFormula sheet →

f^{-1}\big(f(x)\big) = x

$f^{-1}$: the inverse of f, which exists only where f is one-to-one

Use it when you are given a demand or supply function solved for one variable and need it solved for the other.

Exam practice

Summary and review

A function assigns exactly one output to each input; the vertical line test checks this on a graph.
Domain is restricted both algebraically (no division by zero, no negative square roots) and economically (no negative output).
$C(x)$ , $R(x)$ , $P(x) = R(x) - C(x)$ are the basic cost, revenue and profit functions.
Piecewise functions apply different rules on different parts of the domain: identify the correct piece before evaluating.
Composition $(f \circ g)(x) = f(g(x))$ applies the inner function first.
The inverse $f^{-1}$ undoes $f$ ; solving a demand function for $x$ instead of $p$ gives its inverse.