FunctionSurjective

True if every real y is f(x) for some x on f's domain, False otherwise.

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

FunctionSurjective(f, x, Reals)whether f, as a function of the real variable x, is onto the real numbers.

Typed signatures

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

Domain: Elementary functions

Details

• Only the codomain Reals (or RealNumbers) is supported -- any other third argument, or none, declines rather than guess at a codomain this file has no notion of.
• An odd-degree polynomial and Ln (of an affine argument) are onto R; an even-degree polynomial, Exp, Sqrt, and a nonzero constant over an affine denominator (never zero) are not; Tan is onto R.

Examples

1 more kept as data, checked against the oracles

Basic #

An odd-degree polynomial hits every real value, by the intermediate value theorem

live ↗

In$\mathrm{FunctionSurjective}(x^3, x, \mathrm{Reals})$Out$\mathrm{True}$

Known value · Wolfram 15.0.0, hand cross-checked

Exp's range is the positive reals only

live ↗

In$\mathrm{FunctionSurjective}(e^{x}, x, \mathrm{Reals})$Out$\mathrm{False}$

Known value · Wolfram 15.0.0, hand cross-checked

Log is onto R even though its domain is only the positive reals

live ↗

In$\mathrm{FunctionSurjective}(\ln(x), x, \mathrm{Reals})$Out$\mathrm{True}$

A reciprocal never takes the value 0

live ↗

In$\mathrm{FunctionSurjective}(\frac{1}{x}, x, \mathrm{Reals})$Out$\mathrm{False}$

Tangent hits every real value

live ↗

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

Scope #

A cubic is onto the reals

live ↗

In$\mathrm{FunctionSurjective}(x^3-x, x)$Out$\mathrm{True}$

Applications #

The square root never reaches negatives, so not onto

live ↗

In$\mathrm{FunctionSurjective}(\sqrt{x}, x)$Out$\mathrm{False}$

Sine is confined to $[-1,1]$, so not onto

live ↗

In$\mathrm{FunctionSurjective}(\sin(x), x)$Out$\mathrm{False}$

Properties #

A quadratic is bounded below, so not onto

live ↗

In$\mathrm{FunctionSurjective}(x^2+x+1, x)$Out$\mathrm{False}$

An odd-degree polynomial is onto the reals

live ↗

In$\mathrm{FunctionSurjective}(x^3+x+1, x)$Out$\mathrm{True}$

About

Examples	11
Checked against	Wolfram Language 11 agree
Crosswalk	Wolfram Language