Real Numbers, Functions, and Graphs

Real Numbers

Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. The real numbers include all the rational numbers, such as the integer -5 and the fraction $\displaystyle \frac{4}{3}$, and all the irrational numbers such as $\sqrt{2}$ (1.41421356… the square root of two, an irrational algebraic number) and $\pi$ (3.14159265…, a transcendental number).

Any real number can be determined by a possibly infinite decimal representation such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and correspondingly, complex numbers include real numbers as a special case.

Functions

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number $x$ to its square: $f(x)= x^{2}$. Here, the domain is the entire set of real numbers and the function maps each real number to its square. The output of a function $f$ corresponding to an input $x$ is denoted by $f(x)$(read "f of x"). In this example, if the input is $-3$, then the output is 9, and we may write $f(-3) = 9$. The input variable(s) are sometimes referred to as the argument(s) of the function.

Graphs

The graph of a function $f$ is the collection of all ordered pairs $(x, f(x))$. In particular, if $x$ is a real number, "graph" means the graphical representation of this collection, in the form of a line chart, a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching. If the function input $x$ is an ordered pair $(x_1, x_2)$ of real numbers, the graph is the collection of all ordered triples $(x_1, x_2, f(x_1, x_2))$ and its graphical representation is a surface (see the three-dimensional graph below).

Graph of a Function

This is a graph of the function $f(x,y) = \sin{x^2} \cos{y^2}$