Fixed Point — Definition, Formula & Examples
A fixed point of a function is an input value that maps to itself — meaning when you plug it in, you get the same value back out. In other words, is a fixed point of if .
Given a function , a point is called a fixed point of if . Fixed points correspond to the intersections of the graph of with the line .
Key Formula
Where:
- = A function mapping a set to itself
- = The fixed point — the value unchanged by the function
How It Works
To find the fixed points of a function, set and solve for . Graphically, fixed points are where the curve crosses the diagonal line . A function may have zero, one, or many fixed points. In iterative methods, repeatedly applying can sometimes converge to a fixed point, which is the basis of fixed-point iteration in numerical analysis.
Worked Example
Problem: Find the fixed points of f(x) = x² − 2.
Set up the equation: A fixed point satisfies f(x) = x, so set x² − 2 = x.
Rearrange and factor: Move all terms to one side and factor the quadratic.
Solve: Set each factor equal to zero.
Answer: The fixed points are x = 2 and x = −1. You can verify: f(2) = 4 − 2 = 2 and f(−1) = 1 − 2 = −1.
Why It Matters
Fixed-point theory underlies key results in mathematics, including the Banach fixed-point theorem used to prove existence and uniqueness of solutions to differential equations. In numerical analysis, fixed-point iteration is a standard technique for approximating roots of equations. The concept also appears in economics (equilibrium prices) and computer science (recursive definitions and semantics of programming languages).
Common Mistakes
Mistake: Confusing a fixed point with a zero (root) of the function.
Correction: A zero satisfies f(x) = 0, while a fixed point satisfies f(x) = x. These coincide only when x = 0 happens to be both.