Testing Solutions Calculator

x=? — Equation Test

Substitute x₀ into LHS and RHS; check LHS=RHS

f▷c — Inequality Test

Evaluate f(x₀) and g(x₀); check >, ≥, <, ≤, =, ≠

Σ✓ — System Test

Check (x₀,y₀) in every equation; ✓ only if ALL satisfied

P∈? — Point on Curve

Explicit: f(x₀)=y₀? Implicit: F(x₀,y₀)=0?

✓/✗ — Extraneous Filter

Re-test each candidate in original equation; reject failures

∀x — Identity Check

Test at 12 diverse x-values; counterexample → conditional

p(r) — Polynomial Root

p(r)=0? Factor Theorem + synthetic division remainder

f∘g — Composition Test

Eval g(x₀), then f(g(x₀)), compare with h(x₀)

Substitution \(x=x_0\Rightarrow\text{evaluate both sides}\)

Solution def \(f(x_0)=g(x_0)\Rightarrow x_0\text{ is a solution}\)

Inequality \(f(x_0)\operatorname{op}g(x_0)\Rightarrow x_0\text{ in set}\)

System sol \(\forall\text{ eqs satisfied}\Leftrightarrow\text{full solution}\)

On curve \(f(x_0)=y_0\Leftrightarrow(x_0,y_0)\text{ on }y=f(x)\)

Implicit \(F(x_0,y_0)=0\Leftrightarrow\text{on curve}\)

Extraneous \(p(x_0)=0\text{ after squaring}\not\Rightarrow\text{valid}\)

Factor Thm \(p(r)=0\Leftrightarrow(x-r)|p(x)\)

Identity \(f(x)=g(x)\;\forall x\text{ in domain}\)

Conditional \(f(x)=g(x)\text{ for some }x\text{ only}\)

Composition \((f\circ g)(x)=f(g(x))\)

Not commut. \(f\circ g\neq g\circ f\text{ in general}\)

Sqrt extr. \(\sqrt{x}\geq0\;\forall x\geq0\)

Log domain \(\log\text{ requires argument}>0\)

Denom excl. \(q(x_0)=0\Rightarrow x_0\text{ excluded}\)

IVT \(f(a)f(b)<0\Rightarrow\exists\text{ root in }(a,b)\)

✓✗Testing Solutions Calculator