2.7 Composition of Functions

What is composition?

Composition of functions combines two functions so the output of one becomes the input of the other. Written (f∘g)(x) = f(g(x)), you first apply g to x, then apply f to that result. Order matters: f(g(x)) usually differs from g(f(x)). The domain of the composite is restricted to x-values where g(x) exists AND f(g(x)) exists.

Building and decomposing composites

Composite functions can model real situations, like converting units through a chain of functions or modeling a quantity that depends on a rate that itself depends on time. You can also decompose a complicated function into simpler inner and outer functions, which helps with differentiation and understanding structure — e.g., h(x) = √(x²+1) decomposes into f(x) = √x (outer) and g(x) = x²+1 (inner).

Domain restrictions

The domain of f∘g consists of all x in the domain of g such that g(x) is in the domain of f. Even if f and g individually have large domains, composing them can shrink the domain significantly — always check both restrictions rather than assuming the composite's domain matches either original function.

Key terms

Composition of functions

(f∘g)(x) = f(g(x)): apply g first, then apply f to the result.

Is composition commutative?

No — generally f(g(x)) ≠ g(f(x)); order matters.

Domain of f∘g

All x in the domain of g such that g(x) is also in the domain of f.

Decomposing a function

Breaking a complex function into an outer function f and inner function g, such that h(x) = f(g(x)).

Example: f(x)=x², g(x)=x+1, find f(g(3))

g(3) = 4, then f(4) = 16.

Try an AP-style question

f(x) = x² and g(x) = x + 3. What is f(g(2))?

1. 25
2. 13
3. 10
4. 7