tukifunktionax

Tukifunktionax is presented here as a constructed example used in mathematical pedagogy to illustrate a simple saturating, odd function with a tunable oscillatory component. For a fixed order n ∈ N+, the canonical Tukifunktionax f_n: R → R is defined by

f_n(x) = x / (1 + |x|) + sin(n x) / (1 + x^2).

Properties of Tukifunktionax include continuity on the real line and differentiability for all real x, since both terms in the definition are smooth. The function is odd, because each term is an odd function: f_n(-x) = -f_n(x). As |x| grows large, the first term tends to the saturating values ±1, while the second term decays to 0, so f_n(x) approaches ±1 and remains bounded. The first term provides a baseline saturation, and the second term introduces high-frequency, diminishing oscillations whose frequency is controlled by n. This makes f_n a straightforward example of a non-polynomial, bounded, differentiable function with tunable oscillatory behavior.

Origins and usage: The name tukifunktionax is a neologism created for instructional purposes and is not a standard object in reference works. The construction is intended to aid discussion of topics such as saturation behavior, symmetry, and the impact of adding oscillatory components to nonlinear bases. By varying n, students can observe how the graph, derivative, and local extrema evolve with frequency, offering a compact platform for exploring qualitative function properties.

Variants and generalizations: One may modify the saturation term, replace the sine with other periodic functions, or introduce extra parameters to adjust amplitude and decay. Related concepts include saturating activation functions such as tanh and the logistic function.