Porisma
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Chapter exploration

When does an inverse exist?

Explore

Drag either end of the domain. A piece that stays on one side of zero has an inverse; a piece that crosses zero does not.

Choose an interval on the parabola

Move either end. Can two allowed inputs still share an output?

Explore f(x) = x² for a ≤ x ≤ b. An inverse exists when the interval stays on one side of zero.

Domain from a to b-1 to 1

Closed endpoint: the boundary is included unless the original domain excludes it.

Try an example

Not one-to-one: at least one reached output comes from two retained inputs.

Proposed input domain

[-1, 1]

Proposed input domain-303

Retained inputs

[-1, 1]

Retained inputs-303

Reached outputs

[0, 1]

Reached outputs04.59

Original allowed inputs

(−∞, ∞)

These exclusions stay in force when a boundary changes. Arrows indicate continuation beyond the drawing window.

The interval contains inputs on both sides of zero, so some outputs come from two allowed inputs.

The graph shows a finite window; arrows and interval notation describe the full set. Decimal endpoints are rounded.

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