Functions: Notations and Definitions. An “ONTO” Function ONTONOT ONTO (Here: A=Domain, B=Range.)

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Presentation transcript:

Functions: Notations and Definitions

An “ONTO” Function ONTONOT ONTO (Here: A=Domain, B=Range.)

A “ONE to ONE” Function One to one Not one to one. (Here: A=Domain, B=Range.)

How can we find the domains of functions?

How can we find the domains of functions? (continued)

Finding the Ranges of Functions

Find the Domains and Ranges of the following Functions.

More Notation

General Properties of Functions

Looking at Discontinuities

General Properties of Functions

Over Which Intervals are these Functions Increasing, Decreasing or Constant?

General Properties of Functions

Boundedness

General Properties of Functions

Local and Absolute Extrema

General Properties of Functions

Symmetry

General Properties of Functions

Asymptotes

Twelve Basic Functions See Figures 1.36 – 1.47 Pages

Piecewise Functions

Building Functions from Functions

Examples

Composition of Functions

Examples

More Examples

One More Example 3)A store offers a 15% discount on all items and a 20% discount to store employees. a)Write a model for the price found by taking off the 15% discount before the 20% discount. b)Write a model for the price found by taking off the 20% discount before the 15% discount. c)Which results in a cheaper price?

Defining Relations and Functions Implicitly

Defining Relations and Functions Parametrically

Another Example

Inverse Relations and Functions

Finding Inverse Functions

Modeling with Functions We can solve practical problems by modeling them with functions. 1)A parabolic satellite dish with maximum diameter of 24 inches and height of 6 inches is packaged with a cardboard cylinder lodges inside it for protective support. The diameter had a diameter of 12 inches. How high must it be to sit flush with the top of the dish?

More Modeling with Functions 2)Grain leaks through a hole in the bottom of a suspended storage bin at 8 cubic inches per minute. The leaking grain forms a cone whose height is always equal to its radius. If the height is 1 foot tall at 2:00 p.m., how tall will it be at 3:00 p.m.? 3)A car with tires that are 15 inches in radius moves at 70 miles per hour. How many rotations are made per second by the tires?

Graphical Transformations of Functions Two Types of Transformations:  Rigid: size and shape of graph are preserved. (Ex: translations, reflections, rotations)  Nonrigid: size and shape can change. (Scaling, vertical and horizontal stretching and shrinking.)

Translations

Reflections

Reflections of Even and Odd Functions What happens when we reflect even functions across the:  X-axis  Y-axis  Origin  Same question for odd functions.

Stretching or Shrinking Graphs (Scaling)

Combinations of Transformations