Functions can not be seen

Rainer KaendersUniversity of Cologne

GeoGebra Conference Linz 2011

Functions can not be seen

• … but can be represented • GeoGebra can make the identification of

functions with their graphs stronger• Many different representations: show that

there is not the representation of a function• Each representation has ist own possibilities

and failures.• GeoGebra as catalyst for mathematics!

Functions can not be seen

Nomograms

Examples

Van Dormolen [Dor78], Malle (z.B. in [Mal93], S. 265 oder [Mal00], Spivak ([Spi67], S. 79, www.dynagraph.de Hans-Jürgen Elschenbroich

Examples

Examples

Examples

Examples

Examples

Composition

Composition

• Identical Function has a natural appearance• The increase / decrease from x to f(x) becomes visible • Inverse function easy to construct• Involutions, f(x) with f f = id, • Projections, p(x) with p p = p• Composition and Decomposition of two functions • Iterations can be visualized • General notions on mappings of sets (injective,

surjective, sections, …)

Composition

Linear Functions

Linear Functions

Decomposition

Linear Approximation

A Circle as Derivative Curve

Parabola as Derivative Curve

Deltoide and

Run along the Edge

How to use level curves to represent functions?

Run along the Edge

Toblerone Graph

Outlook

r() = C a

Thank you for your attention!

Outline• Functions can not be seen• Nomogrammes

- Composition- Linear Functions- Linear Approximation- Translation of Domain and Value Line- Solving Equations

• Run along the Edge- Toblerone Diagram- Product and Sum of Functions

• Outlook- Various Coordinates