FunctionInjective

True if distinct x on f's domain always give distinct f(x), False otherwise.

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FunctionInjective(f, x)whether f, as a function of the real variable x over its own domain, is one-to-one.

Typed signatures

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

Domain: Elementary functions

Details

Unlike FunctionMonotonicity, injectivity is asked about f's OWN domain, not all of R -- 1/x is injective even though $x\ne0$ .
True for an affine function, an odd-power monomial, Exp, Ln, and a nonzero constant over an affine denominator (a Mobius map); False for an even-power monomial and for Tan (periodic, hence many-to-one on its full domain).
Declines for a quadratic radicand or denominator (symmetric about a vertex in ways this file does not chase down in general) and wherever the domain itself declines.

Examples

Basic #

A reciprocal is injective on its own domain, even though that domain excludes 0

\mathrm{FunctionInjective}(\frac{1}{x}, x)

Out

\mathrm{True}

Known value · Wolfram 15.0.0, hand cross-checked

A cubic is injective (its derivative never goes negative)

\mathrm{FunctionInjective}(x^3, x)

Out

\mathrm{True}

A parabola gives the same value at x and -x

\mathrm{FunctionInjective}(x^2, x)

Out

\mathrm{False}

Known value · Wolfram 15.0.0, hand cross-checked

A nonzero-slope affine function is injective

\mathrm{FunctionInjective}(2x+1, x)

Out

\mathrm{True}

Known value · Wolfram 15.0.0, hand cross-checked

Scope #

The cubic $x^3-x$ has three roots of 0, so not one-to-one

\mathrm{FunctionInjective}(x^3-x, x)

Out

\mathrm{False}

Applications #

Cosine repeats values, so not injective

\mathrm{FunctionInjective}(\cos(x), x)

Out

\mathrm{False}

The square root is one-to-one on its domain

\mathrm{FunctionInjective}(\sqrt{x}, x)

Out

\mathrm{True}

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