FunctionDomain

The set of real x where f (an expression in the single real variable x) is defined, as a condition in x.

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

FunctionDomain(f, x)the domain of f as a function of the real variable x, expressed as an inequality or a disjunction of inequalities.

Typed signatures

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

Domain: Elementary functions

Details

• Built on one shared classifier for an elementary real expression in a single variable — see FunctionMonotonicity, FunctionRange, and the rest of the Function* family, which all reduce the expression the same way before answering their own question.
• Covers: polynomials and rational functions (division by zero, denominator degree <= 2 or a bare power like $x^n$); Sqrt of an affine or quadratic argument (an even root, requiring the radicand >= 0); Ln/Log of an affine argument; and Exp, Sin, Cos, Tan of an affine argument (domain all of $\mathbb R$, except Tan -- see below).
• Declines (leaves the call unevaluated) rather than answer: a second free variable; a denominator or radicand of degree above 2 with more than one term; Log of anything but an affine argument; and Tan, whose true domain excludes a countably infinite set of points ($x \ne \pi/2 + k\pi$) that this file's interval vocabulary cannot express as a finite union of intervals.

Examples

3 more kept as data, checked against the oracles

Basic #

A polynomial is defined everywhere

live ↗

In$\mathrm{FunctionDomain}(x^2, x)$Out$\mathrm{True}$

Known value · Wolfram 15.0.0, hand cross-checked

Division by zero excludes the origin

live ↗

In$\mathrm{FunctionDomain}(\frac{1}{x}, x)$Out$x\lt0∨0\lt x$

Known value · Wolfram 15.0.0, hand cross-checked

Scope #

A rational function with two real poles

live ↗

In$\mathrm{FunctionDomain}(\frac{1}{x^2-1}, x)$Out$x\lt-1∨-1\lt x\lt1∨1\lt x$

Known value · Wolfram 15.0.0, hand cross-checked

An even root needs a nonnegative radicand

live ↗

In$\mathrm{FunctionDomain}(\sqrt{x^2-1}, x)$Out$x≤-1∨1≤x$

Known value · Wolfram 15.0.0, hand cross-checked

A natural log needs a positive argument

live ↗

In$\mathrm{FunctionDomain}(\ln(2-x), x)$Out$x\lt2$

Known value · Wolfram 15.0.0, hand cross-checked

Possible issues #

Tan's excluded set isn't a finite union of intervals, so this stays unevaluated rather than guess

live ↗

In$\mathrm{FunctionDomain}(\tan(x), x)$Out$\mathrm{FunctionDomain}(\tan(x), x)$

Known value · Wolfram 15.0.0, hand cross-checked

About

Examples	9
Checked against	Wolfram Language 7 agree2 differ
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