One input per output, or no inverse: An inverse exists only when the function is one-to-one (passes the horizontal-line test).
No inverse? Restrict the domain
- If two x's give the same y, there's no single way back.
- Fix it by restricting the domain to a stretch where the function is one-to-one.
- Example: f(x) = x² needs x ≥ 0.
IB-style question — restrict, then invert
f(x) = x² is not one-to-one on ℝ.
Restrict its domain so f⁻¹ exists, and find f⁻¹.
Step by step
- Restrict to a half where it's one-to-one, say x ≥ 0.
- Swap x and y and solve: y = x² ⇒ x = √y (positive root).
Final answer
Restrict to x ≥ 0; then f⁻¹(x) = √x.
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Its own inverse: A self-inverse function is its own inverse: f(f(x)) = x, so f⁻¹ = f. Its graph is symmetric in the line y = x.
Classic examples: f(x) = 1/x and f(x) = a − x.
IB-style question — show self-inverse
Show that f(x) = (2x + 3)/(x − 2) is self-inverse.
Step by step
- Find f⁻¹: write y = (2x + 3)/(x − 2) and make x the subject.
- Solve for x.
- Swap to get f⁻¹(x) — it's the same rule as f.
Final answer
f⁻¹ = f, so f is self-inverse.