FunctionRange

The set of values f takes as x ranges over its domain, as a condition in y.

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FunctionRange(f, x, y)the range of f as a function of the real variable x, expressed as a condition in y.

Typed signatures

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

Domain: Elementary functions

Details

Uses the same classifier as FunctionDomain. An odd-degree polynomial is onto $\mathbb R$ by the intermediate value theorem regardless of its other coefficients; a degree-2 polynomial's range comes from its vertex.
Covers: odd-degree polynomials (True); degree-2 polynomials (from the vertex); a nonzero constant over an affine denominator ( $y \ne 0$ ); Sqrt of an affine or quadratic argument; Ln of an affine argument (True); Exp of an affine argument ( $y>0$ ); Sin/Cos of an affine argument ( $-1\le y\le1$ ); Tan of an affine argument (True).
Declines for even-degree polynomials above degree 2 (no general closed-form extremum here), and for a rational function whose denominator is a quadratic (needs the same vertex analysis compounded with the reciprocal, left for later).

Examples

Basic #

A parabola's range is bounded below by its vertex

\mathrm{FunctionRange}(x^2, x, y)

Out

0≤y

Known value · Wolfram 15.0.0, hand cross-checked

A reciprocal never reaches zero

\mathrm{FunctionRange}(\frac{1}{x}, x, y)

Out

y\lt0∨0\lt y

Sine's range is the unit interval

\mathrm{FunctionRange}(\sin(x), x, y)

Out

-1≤y≤1

Known value · Wolfram 15.0.0, hand cross-checked

Scope #

Exp's range is the positive reals

\mathrm{FunctionRange}(e^{x}, x, y)

Out

0\lt y

Known value · Wolfram 15.0.0, hand cross-checked

A parabola is bounded below by its vertex value $-1$

\mathrm{FunctionRange}(x^2-1, x, y)

Out

-1≤y

Applications #

The vertex gives the minimum $-4$

\mathrm{FunctionRange}(x^2-4, x, y)

Out

-4≤y

Tangent takes every real value

\mathrm{FunctionRange}(\tan(x), x, y)

Out

\mathrm{True}

Possible issues #

A quadratic denominator's range needs its vertex value too, and stays unevaluated for now

\mathrm{FunctionRange}(\frac{1}{x^2-1}, x, y)

Out

\mathrm{FunctionRange}(\frac{1}{x^2-1}, x, y)

Known value · Wolfram 15.0.0, hand cross-checked

About

Examples	8
Checked against	Wolfram Language 7 agree1 differ
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