Introducing Functions

    Descarte's Cartesian coordinate system is defined and then it is shown how to make graphs of relations. Functions are then defined and it is explained how to determine if a set or ordered pairs defines a function or not. The vertical line test is detailed and functional evaluation is shown.

Definition (Cartesian Coordinate System) In order to represent points in a plane by pairs of real numbers, we select two intersecting perpendicular lines and establish a scale on each line. The point of intersection is called the origin. These two lines, the axes, are distinguished by labeling with and .  For a given point in the plane there corresponds an -coordinate and a -coordinate, called the abcissa and the ordinate, respectively.

    There is a one-to-one correspondence between the points in a plane and the elements in the set of all ordered pairs of real numbers.

Definition (Relation) A relation is any set of ordered pairs; that is a relationship between a first set (called the domain) and the second set (called the range) such that each member of the domain corresponds to at least one member of the range. The domain is sometimes called the set of inputs and the range sometimes called the set of outputs.

Example (Relation) (a) The set does not define a relation because 1,2 and 3 are not ordered pairs.
    (b) The set


    
defines a relation because each of and are ordered pairs of real numbers. We sometimes write and

Definition (Function) A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range. Elements of the domain are sometimes called inputs and elements of the range are sometimes called outputs.

Example (Function) (a) The set



does not define a function because the input of has two different outputs of and
    (b) The set does define a function because every input has a unique output. Namely and

Definition (Vertical Line Test) If it is possible for a vertical line to cross the graph more than once, then the graph is not the graph of a function.

Example (Vertical Line Test) (a) The set



does not define a function. Sketching the graph of these points it is possible to find a vertical line that passes through the points and So does not define a function.
     (b) The set of points that satifies the equation does define a function. From the graph
     
     
we can see that there are no vertical lines that pass through the graph more than once.

Definition (Functional Notation) For any element in the domain of the function, the symbol represents the element in the range of corresponding to in the domain of If is an input value, then is the corrsponding output value. If is an element that is not in the domain of then is not defined at and does not exist.

Example (Functional Notation) Evaluate for and where is defined. We have

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Cite this as:
Introducing Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/introducing-functions.html