Explore form-preserving functions

Use .ff for both preserved polynomial forms and fractional forms. Values are ordinary callables.

Runnable RiXRun cell

.Plugin.Load("fracfun"); P := .ff`(x+1)*(x+1)`; R := .ff`(x^2-1)/(x-1)`; Q := (`x^2+1`).ff(); {: P(2), R(3), Q(2), P.Form(), R.Form() };

Operations preserve their construction

Addition and multiplication retain their expression trees. Together is an explicit choice.

Runnable RiXRun cell

.Plugin.Load("fracfun"); A := .ff`1/x`; B := .ff`1/(x+1)`; Sum := A+B; Product := A*B; {: Sum.Form(), Product.Form(), Sum.Together().Form() };

Expand and recenter explicitly

Runnable RiXRun cell

.Plugin.Load("fracfun"); P := .ff`(x+1)*(x+1)`; Q := .ff`x^2+1`; {: P.Expand().Form(), Q.Recenter(2).Form() };

Compare form, value, and function domain

Runnable RiXRun cell

.Plugin.Load("fracfun"); A := .ff`(x^2-1)/(x-1)`; B := .ff`x+1`; C := A.Cancel(); {: A == B, A.Equivalent(B), A.SameFunction(B), C.Equivalent(B), C.SameFunction(A) };

C displays a cancelled form but still has the source hole at 1. Crossing to canonical RationalFunction semantics is explicit.

Runnable RiXRun cell

.Plugin.Load("fracfun"); F := .ff`(x^2-1)/(x-1)`; C := F.Cancel(); {: F.Domain(), C.Domain(), F.Canonical()(1) };

Calling C(1) still reports division by zero, while F.Canonical()(1) is 2.

Canonical projections

Runnable RiXRun cell

.Plugin.Load("fracfun"); PForm := .ff`(x+1)*(x+1)`; RForm := .ff`(x^2-1)/(x-1)`; P := PForm.P(); R := RForm.R(); {: P(3), R(3), PForm.Record(), RForm.Record() };

Verified presentations that retain the source domain

Factorization, square-free decomposition, and partial fractions operate on the canonical exact projection. Their FractionFunction wrappers keep the original domain authoritative:

Runnable RiXRun cell

.Plugin.Load("fracfun"); F := .ff`(x^2-1)/(x-1)`; factored := F.Factor(); squareFree := F.SquareFree(); partial := F.PartialFractions(); {: factored[:verified], factored.Get("sourceDomainPreserved"), factored.Presentation(), squareFree.Presentation(), partial.Presentation(), factored.Domain() };

factored.Source()(1) still fails because the source denominator is zero. factored.Canonical()(1) is 2; that is an explicitly requested canonical projection, not a widening of the source function's domain.

Zeros, poles, and removable holes

Runnable RiXRun cell

.Plugin.Load("fracfun"); F := .ff`(x^2-1)/(x-1)`; divisors := F.PoleZeroEvidence(); holes := F.RemovableHoleEvidence(); {: divisors[:zeros], divisors[:poles], divisors.Get("removableHoles"), holes.Get("cancelledFactorEvidence"), holes.Get("restrictionEvidence"), holes[:complete] };

The canonical zero is -1, there are no canonical poles, and the cancelled source restriction gives the removable hole at 1. When an irreducible factor remains, verified still certifies the displayed algebra while complete stays null and the residual polynomial records what is not yet enumerated.

For a classroom comparison, render all three transformations together or one at a time:

Runnable RiXRun cell

.Plugin.Load("fracfun"); F := .ff`(x^2+1)/(x^2-1)`; F.TransformationGrid(); F.Factored().Grid();

Export forms and restrictions to Calculus

.symbolic loads FractionFunction and Calculus together. Displayed and source-domain evaluation forms cross the public expression boundary separately, so cancellation cannot hide a removable hole:

Runnable RiXRun cell

.Plugin.Load("symbolic"); F := .ff`(x^2-1)/(x-1)`; C := F.Cancel(); {: .calculus.ToSpec(C.CalculusExpression()), .calculus.ToSpec(C.EvaluationCalculusExpression()), C.RestrictionExpressions(), .symbolic.Obligations(C) };

Differentiate the exported form with the obligation-bearing Calculus result:

Runnable RiXRun cell

.Plugin.Load("symbolic"); F := .ff`(x^2-1)/(x-1)`; result := .symbolic.DifferentiateResult(F, :x); {: .calculus.ToSpec(result[:expression]), result[:obligations], F.Domain().Get("calculusRestrictions") };

The derivative result retains the quotient's nonzero denominator condition. .symbolic.Obligations(F) exposes the same source-domain restriction directly from FractionFunction metadata.

Polynomial FractionFunctions also use the higher-derivative and evaluation façades without leaving the public Calculus representation:

Runnable RiXRun cell

.Plugin.Load("symbolic"); P := .ff`x^3+x`; second := .symbolic.DifferentiateN(P,:x,2); {: .calculus.ToSpec(second), .symbolic.Evaluate(second,{= x=3 }) };

They can also formulate integral records without erasing source-domain holes:

Runnable RiXRun cell

.Plugin.Load("symbolic"); F := .ff`1/x`; x := .calculus.Variable(:x); family := .symbolic.AntiderivativeFamily(F,:x,.calculus.Log()(x),:C); definite := .symbolic.DefiniteIntegral(F,:x,1,2); {: family[:representative], family[:constant], family[:obligations], definite[:lower], definite[:upper], definite[:obligations] };

Both records inherit x != 0 from the FractionFunction. The family merely selects Log(x) as a representative plus an explicit constant; it does not silently certify the identity or discharge the restriction.

Reactive forms

Runnable RiXRun cell

.Plugin.Load("fracfun"); $$y := 2; $$F := `.ff.Var(x):(x+@($y))/(x-1)`; $$G := $F + .ff`1/(x-1)`; $y := 4; {: $F(3), $G(3), $G.Form() };