Algebraic Symmetry Suite
Inverse Function Solver
Reverse your functions with ease. Swap the roles of X and Y instantly with our specialized algebraic engine.
Inverse Function Solver
Original Function: f(x) = ax + b
Inverse Functionf⁻¹(x) = 0.5x - 2
One-to-One Property
A function has an inverse only if it is "one-to-one" (injective), meaning it passes the horizontal line test. For a linear function, this simply means the slope a must not be zero.
The Concept
An inverse function essentially "undoes" the original function. If f(x) maps A to B, then f⁻¹(x) maps B back to A.
Graphing Reflection
The graph of an inverse function is the reflection of the original function's graph across the line y = x.
Key Properties
- Domain Swap: The domain of f becomes the range of f⁻¹.
- Symmetry: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
- Existence: Only functions that are one-to-one have inverses.
- Notation: f⁻¹(x) does NOT mean 1/f(x).
Quick Check
A vertical line test checks if a relation is a function; a horizontal line test checks if a function has an inverse.