INVERSE FUNCTIONS: UNDO
PAGE 1 OF 5, WHAT IS f⁻¹? THE UNDO BUTTON
INVERSE f⁻¹
THE FUNCTION THAT UNDOES f
If a function f sends an input x to an output y, its inverse f⁻¹ sends y back to x. Think of f as a machine and f⁻¹ as the undo button. The defining identities are f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, whenever both sides make sense. The notation f⁻¹(x) means "the inverse function of f," not 1/f(x). A reciprocal divides; an inverse reverses. Every invertible function has exactly one inverse partner, and together they form a matched pair that cancels itself out.
⚡ KEY RULE
f⁻¹ undoes f: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. f⁻¹(x) is not the same as 1/f(x).
UNDO!
NOTATION
f⁻¹(x) ≠ 1/f(x)
🚫 Not a reciprocal
✅ Means "undo f"
🚫 Not a reciprocal
✅ Means "undo f"
PAIR
f ↔ f⁻¹ partners
🔄 Input becomes output
↩️ Output goes back
🔄 Input becomes output
↩️ Output goes back