Function Composition Calculator
Enter f(x), g(x), and x0 to compose functions, compare order, and see each evaluation step.
Nested Mapping Topologies 🖖
Function composition strictly cascades mathematical operations, passing the codomain output of the inner function as the precise domain input for the outer. This deeply recursive mapping fundamentally exposes how non-commutative chained operations construct deeply complex analytical surfaces.
Two machines in a row 🖖
Composing functions just means running them back to back: f(g(x)) feeds x into g first, then pours g's answer into f. Think of an assembly line where each station reshapes the part before passing it on. The practical takeaway: always evaluate from the innermost parentheses outward, so g(x0) comes before f.
The fractal hiding in repeated composition 🖖
Compose one simple function with itself over and over and you leave algebra for chaos theory. Iterating z ā¦ zĀ² + c ā the same machine feeding its own output back in ā is the exact rule that draws the Mandelbrot set: a point belongs only if that endless self-composition stays bounded. So the humble f(g(x)) button is one step away from generating infinitely detailed fractals.
Example problems
- linear after quadratic - Step-by-step f(g(x)) and g(f(x)) for linear and quadratic functions.
- quadratic after linear - Shows why reversing composition order changes the expression and value.
- square after sine - square after sine
- domain example - Domain restriction example where the inner output can make a composition undefined.
- same at one point - samePoint sample case