Functions and domains
A real-valued function $f$ of a real-valued variable $x$ is a rule assigns to each real number $x$ in a specified set of numbers, called the domain of $f,$ a single real number $f(x),$ read '$f$ of $x.$'The quantity $x$ is called the argument of $f$ and $f(x)$ is called the value of $f$ at $x.$
A function is usually specified

numerically using a table of values, graphically using a graph, or algebraically using a formula,

and also in other ways.
Note on domains The domain of a function is not always specified explicitly; if no domain is specified for the function $f,$ we take the domain to be the largest set of numbers $x$ for which $f(x)$ makes sense. This 'largest possible domain' is sometimes called the natural domain.UtilitiesFunction evaluator and grapher: Evaluate and graph functions. Excel grapher: Download an Excel® sheet that graphs functions.

Examples: Functions and domainsA function specified numerically:A function specified graphically:A function specified algebraically:

Intervals
The closed interval $[a,b]$ is the set of all real numbers $x$ with $a \leq x \leq b.$
The open interval $(a,b)$ is the set of all real numbers $x$ with $a \lt x \lt b.$
The interval $(a,+\infty)$ is the set of all real numbers $x$ with $a \lt x \lt +\infty.$
The interval $(-\infty,b)$ is the set of all real numbers $x$ with $-\infty \lt x \lt b.$
We also have half open intervals of the form$[a, b)$ and $(a, b].$

Example: Intervals

Graph of a function
The graph of a function $f$is the set of all points of the form $(a, f(a))$ in the $xy$-plane, where we restrict the values of $a$ to lie in the domain of $f.$The following diagram illustrates the graph of a function

Vertical Line TestFor a graph to be the graph of a function, each vertical line must intersect the graph in at most one point.

Examples: Graph of a functionSketching a graph: To sketch the graph of

$f(x)=2x^2-3x+1 \qquad$ Function form

with domain $[0,+\infty),$ we replace $f(x)$ by $y,$ getting the equation

$y=2x^2-3x+1 \qquad \quad \ $ Equation form

We then graph it by plotting points, restricting $x$ to lie in $[0,+\infty),$ and obtain the following picture:

Matching functions to their graphs

Piecewise defined functions
It is sometime necessary to use two or more formulas to specify a single function algebraically. A piecewise defined function is a function whose algebraic definition changes according to the value of the argument. For instance, a piecewise defined function with three formulas might look like this:

$\displaystyle f(x) = \begin{cases} p(x) &\text{ if } x \lt a\\q(x) &\text{ if } a \leq x \leq b\\r(x) &\text{ if } x \gt b\end{cases}$

in which case its graph would have the following form:

$y=p(x)$$y=q(x)$$y=r(x)$

Example: Piecewise defined functions

Function and equation notation
Sometimes, intead of writing, say,

$f(x) = 5x^2-4x+1$

Function notation

we can write

$y = 5x^2-4x+1$

Equation notation

or perhaps

$f = 5x^2-4x+1.$

We can use any letter.

In an equation of the form $y=$ Expression in $x$, we call the argument $x$ the independent variable, and $y$ the dependent variable (because the value of $y$ depends on a choice of the value for $x.$

Example: Function and equation notation
We can think of the equation $C = -55x^2 - 8x$ in two ways:

1. As an equation with independent variable $x$ and dependent variable $C.$
2. As specifying a function:
$C(x) = -55x^2 - 8x$

so we sometimes say that '$C$ is a function of $x$.'

Mathematical models
To mathematically model a situation means to represent it in mathematical terms. The particular representation used is called a mathematical model of the situation.
Types of modelsAnalytical models are obtained by analyzing the situation being modeled, Curve-fitting models are obtained by using mathematical formulas to approximate observed data.

Examples: Mathematical modelsAnalytical models:The temperature on Mars is now −80°F and increasing by 20°F per hour.

Model: $T(t) = -80 + 20t $ ($t$ = time in hours, $T$ = temperature)

Curve-fitting model:

Cost, revenue and profit models
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 $m,$ the marginal cost, measures the incremental cost per item.
A revenue function $R$ gives the total revenue $R(x)$ from the sale of $x$ items.
If the price is fixed at \$$k$ per item, then

Revenue = Price × quantity $R(x) = kx.$

A profit function $P$ gives the total profit $P(x)$ from the sale of $x$ items. The profit, cost and revenue functions are related by the formula

$P(x) = R(x)-C(x)$

Break-even occurs when

$P(x) = 0$

or, equivalently, when

$R(x)=C(x)$

Example: Cost, revenue and profit models

Demand and supply models
A demand equation or demand function expresses demand $q$ (the number of items demanded) as a function of the unit price $p$ (the price per item). A supply equation or supply function expresses supply $q$ (the number of items a supplier is willing to bring to the market) as a function of the unit price $p$ (the price per item). It is usually the case that demand decreases and supply increases as the unit price increases. Demand and supply are said to be in equilibrium when demand equals supply. The corresponding values of $p$ and $q$ are called the equilibrium price and equilibrium demand. To find the equilibrium price, determine the unit price $p$ where the demand and supply curves cross (sometimes we can determine this value analytically by setting demand equal to supply and solving for $p$). To find the equilibrium demand, evaluate the demand (or supply) function at the equilibrium price.

Examples: Demand and supply models
If the demand for Ludington"s Wellington Boots is $q = -4.5p + 4,000$ pairs of boots sold per week and the supply is $q = 50p - 1,995$ pairs per week (see the graph below), then the equilibrium point is obtained when demand = supply:

$-4.5p+4,000 = 50p-1,995$ $54.5p = 5,995$

giving $p = 5995/54.5 = \$110.$The equilibrium price is therefore \$110 and the equilibrium demand is $q=-4.5(110) + 4,000 = 3,505$ pairs per week. What happens at prices other than the equilibrium price can be seen in the following figure:

When the price is lower that the equilibrium price, the demand is greater than the supply, resulting in a shortage.

When the price is set at the equilibrium price, the demand equals supply, so there is no shortage or surplus, and we say that the market clears.

When the price is greater that the equilibrium price, the supply is greater than the demand, resulting in a surplus.