Explanations
How to read the table
Håkan's own explanations, in his own words: what a function is, what the glyphs mean, and how the colours connect the squares.
What is a function?
“A function is a rule f which for each member x in a set Df (domain) orders a specific member f(x) in another set Vf (range).”
For the functions in this table both sets are real numbers only. The graph is the set of points (x, f(x)), and the definition means every vertical line meets the graph at most once.
A square
The same square, calm and fully annotated. Around the graph live the domain and range; below them the special points, asymptotes and generalized integrals; at the bottom the rule of calculation. Every glyph is defined on the Explanations poster.
The seven relations
- Inverse function. f(x) → f⁻¹(x): flip the graph in the diagonal, from lower left to upper right.
- Inverted value. f(x) → 1/f(x): invert vertically in the line y = 1 (if positive) or y = −1 (if negative).
- Derivative. f(x) → f′(x): the value f′(x) is the slope of the curve at x.
- Laplace transform. f(x) → L[f](x) = ∫₀^∞ e^(−xt) f(t) dt.
- Fourier transform. f(x) → F[f](x): the amount of frequency x in f.
- Dilate. f(x) → b·f(ax): compressed or expanded, vertically and/or horizontally.
- Translate. f(x) → f(x−a) + b: moved with no change of shape.
Properties, in plain words
A few of the ~30. Each has its own glyph on the squares. Håkan's glosses, verbatim:
Convexconnected, no right turns
Concaveconnected, no left turns
Oddunchanged if rotated 180° around the centre
Evensymmetric in the vertical central line
Strictly increasinguphill to the right, no plateau
Injectivehorizontal lines have at most one intersection
Surjective on Rhorizontal lines have at least one intersection
Periodica piece of length p constantly repeating