FunctionSign

1 if f is nonnegative over its whole domain, -1 if nonpositive, Indeterminate otherwise.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageFunctionSign!↗

FunctionSign(f, x)the sign of f in the real variable x, when it is constant over the whole real line.

Typed signatures

Type	Package	Applies
(expression, symbol) -> expression	analytic	always

Domain: Elementary functions

Details

• Matches Wolfram's convention that this needs domain $\mathbb R$ to answer definitively -- FunctionSign(Sqrt(x), x) is Indeterminate even though $\sqrt x\ge0$ wherever it's defined, because $\sqrt x$ isn't defined for every real x (confirmed against wolframscript).
• A polynomial's sign comes from its closed-form shape (see FunctionMonotonicity's classifier); a rational function's sign is the product of its numerator's and denominator's, when both are individually determinate; Exp is always 1.

Examples

Basic #

A perfect square is never negative

live ↗

In$\mathrm{FunctionSign}(x^2, x)$Out$1$

Known value · Wolfram 15.0.0, hand cross-checked

1/(x^2+1) is defined and positive for every real x

live ↗

In$\mathrm{FunctionSign}(\frac{1}{x^2+1}, x)$Out$1$

Known value · Wolfram 15.0.0, hand cross-checked

Exp is always positive

live ↗

In$\mathrm{FunctionSign}(e^{x}, x)$Out$1$

Scope #

Squares are never negative, so the sign is 1 on a list

live ↗

In$\mathrm{Map}(\operatorname{\_1}↦\mathrm{FunctionSign}(\operatorname{\_1}, x), \bigl\lbrack x^2, x^2+1\bigr\rbrack)$Out$\bigl\lbrack1, 1\bigr\rbrack$

Properties #

A cubic is negative for x<0 and positive for x>0

live ↗

In$\mathrm{FunctionSign}(x^3, x)$Out$\mathrm{FunctionSign}(x^3, x)$

Known value · Wolfram 15.0.0, hand cross-checked

About

Examples	5
Checked against	Wolfram Language 4 agree1 differ
Crosswalk	Wolfram Language