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2.1 - Functions and Relations

Author: Kyle Webb

What is a function

Function Notation

Difference Ratio

One of the basic definitions in calculus employs the ratio:

fraction numerator f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis over denominator h end fraction

Source: MICDS

Vertical Line Test - Is something a function?

Discrete and Continuous Relations and Functions

A discrete function or relation is one in which the domain can be represented by a finite number of values.  For example, if we consider the relation open curly brackets left parenthesis 2 comma 2 right parenthesis comma left parenthesis 5 comma 1 right parenthesis comma left parenthesis 6 comma 2 right parenthesis comma left parenthesis 9 comma 4 right parenthesis close curly brackets, we see that the graph represents a set of individual points.  This relation can further be classified as a discrete function as it also passes the vertical line test.  

 

A continuous function or relation is one in which the domain contains an infinite number of points and the graph can be represented by a smooth curve or line.  For example, consider the function f left parenthesis x right parenthesis equals 3 x squared minus 1.  The domain consists of all real numbers x, and the graph is shown in the figure below.  

 

 

Source: MICDS