Define functions and build a chain. Press Calculate.
The dot(s) mark the sample input \(x_0\). Their coordinate labels are shown as \((x_0,\,y)\).
Function Composer
Math Algebra • Functions
Frequently Asked Questions
What is function composition f(g(x))?
Function composition means applying g first and then using its output as the input to f. In notation, (f o g)(x) = f(g(x)).
How do I compute f(g(x)) by substitution?
Replace every x in the outer function f(x) with the inner expression g(x), then simplify. For example, if f(x) = x^2 + 1 and g(x) = 2x - 3, then f(g(x)) = (2x - 3)^2 + 1.
Why does the domain change when composing functions?
The input x must be allowed for g, and the output g(x) must also be allowed for f. Restrictions like sqrt input >= 0 or ln input > 0 must be enforced after substitution, which can exclude additional x-values.
Does the order of functions in the chain matter?
Yes, composition is not generally commutative. In most cases f(g(x)) is different from g(f(x)), so reordering the chain changes the result.
How does the chain rule relate to f(g(x))?
If both derivatives exist, the derivative follows (d/dx) f(g(x)) = f'(g(x)) x g'(x). For longer chains, the derivative is a product of derivatives evaluated at the nested inner expressions.