Economics 214 Lecture 3 Introduction to Functions

Variables Variables studied in economics can be qualitative or quantitative. Qualitative variable represents some distinguishing characteristic, such male or female, employed or unemployed. Quantitative variables can be measured numerically.

Numbers Integers are whole numbers. Real numbers include all integers and all numbers between the integers. Numbers that can be expressed as ratios of integers are call rational numbers. Numbers that cannot be expressed as ratios of integers are call irrational numbers.

Intervals Interval is set of all real numbers between to endpoints. Closed interval includes the endpoints. i.e. [1,2] Open interval between two numbers excludes the endpoints. i.e. (1,2) Half-closed or half-open interval between 2 numbers includes one endpoint and excludes the other endpoint. i.e. (1,2] Infinite interval has negative infinity, positive infinity or both as endpoints. i.e. [0,∞)

Sets Set is simply a collection of items. Item included in a set are called elements. C={freshman,sophmore,junior,senior} To show item is part of set we use symbol, . i.e. freshman  C To show item is not part of set, we use symbol, . i.e. graduate student  C.

Sets A set can be described either by listing all its elements or by describing the conditions required for membership. For Example N={10,20,30,40} Or N={x|x=10*y, y=1,2,3,4}

Relations The elements of one set can be associated with the elements of another set through a relationship. A function is a relationship that has a rule that associates each element of one set with a single element of another set. A function is also called a mapping or a transformation.

Function A function f that unambiguously associates with each element of a set X one element in the set Y is written as f:X  Y. The set X is called the domain of the function f. The set of values that occur is called the range of the function f.

Example Function X={1,2,3,4} f:Y=10X Y={10,20,30,40} f:X  Y

Univariate Function A Univariate function maps one number, a member of the domain, to one and only one number, element of the range. We represent the univariate function as y=f(x). Y is the dependent variable or value of the function. x is the independent variable or argument of the function.

Examples of Univariate functions

Ordered Pairs An Ordered pair is two numbers presented in parentheses and separated by a comma, where the first number represents the argument of the function and the second number represents the corresponding value of the function. Each ordered pair for the function y=f(x) takes the form (x,y).

Example of Ordered Pair

Graphing Ordered pairs can be plotted in a Cartesian plane. The origin of the plane occurs at the intersection of the two axes that are a right angles to each other. Points along the horizontal axis represent values of the argument of the function.

Graphing Continued Points along the vertical axis represent values of the function. The coordinates of a point are the values of its ordered pair and represent the address of that point in the plane. The x-coordinate of the pair (x,y) is called the abscissa, and the y-coordinate is called the ordinate. The origin is represented by the ordered pair (0,0).

Plot of our function

Graph Graph of a function represents all points whose coordinates are ordered pairs of the function.

Graph of our function

Linear function

Graph Linear Function

Graph function in multiple quadrants