section 1.2 functions and their properties

Section 1.2 Functions and their Properties

Aaron ThomasJacob WefelTyler Sneen

Funny Introduction

Introduction

By the end of this lesson we will introduce the terminology that is used to describe functions

These include: Domain, Range, Continuity, Discontinuity, upper and lower bound, Local and absolute maximums and minimums, and asymptotes

Domain and Range

The domain of a function is all of the possible x-values the function can have. It can be expressed as an inequality

The Range of a function is all of the possible y-values the function can have. It is also expressed as an inequality

Domain and Range Example

Domain: All Real Numbers Range: All Real Numbers

Domain: x> -1 Range: x>-5

Discontinuity

A graph has continuity if its graph is connected to itself throughout infinity. There are no asymptotes or holes in the graph

A Graph has removable discontinuity if its graph has a hole where one x value was removed from the domain

A graph has infinite discontinuity if its graph has an asymptote that can not be replaced with only one value

Discontinuity Example

Jump Discontinuity Removable Discontinuity

Bindings

A function is bounded above or below if the graph’s range doesn’t extend past a certain point above or below.

A function is “Bounded” if the function’s range doesn’t extend below or above certain points

If the function has no restrictions on its range’s extent the function is considered “unbounded”

Bindings Example

This sine function is bounded above and below at 1 and -1

Max and Mins

A Local Maximum/Minimum of a function is the highest/lowest point of the range in the surrounding window of the graph

The absolute maximum/minimum of a function is the highest/lowest point of the entire range of the graph

Max and Min Example

Local Min: 3, -4, 4 Local Max: 5 Absolute Max: None (Graph goes

infinitely upward) Absolute Min: -4

Asymptotes

A horizontal asymptote is a part of the function which gets infinitely close to a Y-value but never touches it

A Vertical asymptote is a part of the function which gets infinitely close to a x- value but never touches it

Asymptotes Example Identify any horizontal or vertical asymptotes of

the graph of You would first start by foiling the denominator…

= (x+1)(x-2) This means that the graph has vertical

asymptotes of x=-1 and x=2 Because the denominator’s power is bigger than

the numerator’s, y = 0 no matter what the value of x is

Now you have x/((x+1)(x-2)) = 0 This means that the horizontal asymptote is zero