Real functions and economic models

1. Sets and operations

A set is a collection of objects, called elements; $x\in A$ means that $x$ is an element of $A$, and $A\subseteq B$ means that every element of $A$ also belongs to $B$. For two sets $A$ and $B$:

• the union $A\cup B$ consists of the elements that are in $A$ or in $B$;
• the intersection $A\cap B$ consists of the elements that are both in $A$ and in $B$;
• the difference $A\setminus B$ consists of the elements of $A$ that are not in $B$.

The number sets used in the course are $\mathbb{N}\subseteq \mathbb{Z}\subseteq \mathbb{Q}\subseteq \mathbb{R}$, that is, the natural numbers, the integers, the rationals and the reals. The solutions of a system of inequalities form the intersection of the solutions of the individual inequalities, because they must satisfy all of them at the same time. For the complete theory of sets, see the topic on sets in Mathematical Analysis 1.

2. Intervals

An interval is a set of real numbers lying between two endpoints. For $a<b$:

and in the same way $[a,b)$ includes the endpoint $a$ but not $b$, and $(a,b]$ includes $b$ but not $a$. The unbounded intervals are the half-lines, such as $[a,+\infty )=\{x\in \mathbb{R}:x\ge a\}$ and $(-\infty ,b)=\{x\in \mathbb{R}:x<b\}$; the symbols $\pm \infty$ are not numbers, so they never belong to an interval. Quantities produced or sold are not negative, so in economic models the variables usually lie in $[0,+\infty )$.

1. Real function of a real variable

Let $A\subseteq \mathbb{R}$ and $B\subseteq \mathbb{R}$. A function $f:A\to B$ assigns to each $x\in A$ one and only one element $f(x)\in B$. The set $A$ is the domain, $x$ is the independent variable and $f(x)$ is the image of $x$. The set of all images,

is the range of the function, and it is contained in $B$. For example $f(x)=x^2$, defined on $\mathbb{R}$, has range $[0,+\infty )$, because a square is never negative and every number $y\ge 0$ is the square of $\sqrt{y}$.

2. Graph of a function

The graph of $f:A\to \mathbb{R}$ is the set of points of the plane

Since each $x$ corresponds to only one value, every vertical line $x=x_0$ meets the graph in at most one point, precisely in one point if $x_0\in A$ and in none if $x_0\notin A$. For this reason a circle is not the graph of a function $y=f(x)$, whereas its upper semicircle is. The point $(x_0,y_0)$ belongs to the graph if and only if $x_0\in A$ and $y_0=f(x_0)$; with $f(x)=x^2-3$, the point $(2,1)$ belongs to the graph because $2^2-3=1$, the point $(1,1)$ does not because $1^2-3=-2$.