Section 1.2 Function Definitions

Function Definitions

Functions are relations between variables that have one output for every input. Functions can be categorized as one of six different types, each one having a contradictory type.

Increasing and Decreasing Functions

 A function f is increasing on an interval if, for any andin the interval,

implies

Basically, lesser inputs yield lesser outputs.

An example of an increasing function would be when x=y

A function f is decreasing on an interval if, for any andin the interval,

implies

In this case, lesser inputs yield greater outputs.

When compared to an increasing function, decreasing functions can be the same, but with opposite y values.

Maximum and Minimum Values

A function value f (a) is called a relative maximum of f if there exists an interval
that contains a such that 

 implies

A maximum is global when it is the greatest value in the entire graph, and is local when it is the greatest value within a selected portion of the graph.

A function value f (a) is called a relative minimum of f if there exists an interval

that contains a such that 

 implies 

A minimum value can also be global or local, and both minimum and maximum are used for the y values of a graph.

A function which shows a local maximum and minimum, and its global maximum

Even and Odd Functions

A function f is even if, for each x in the domain of f ,

The most noticeable feature of an even function is that it will always have a line of symmetry at x=0. The y values will be equivalent on both sides of the y axis as long as they are equal in terms of horizontal displacement from the y axis.

Some even functions

A function f is odd if, for each x in the domain of f ,

Odd functions are still symmetrical, but are symmetrical about the origin unlike how even functions are symmetrical about the y axis.

An odd function