Mathwords: Composition

Combining two functions by substituting one function's formula in place of each x in the other function's formula. The composition of functions f and g is written f ° g, and is read aloud "f composed with g." The formula for f ° g is written (f ° g)(x). This is read aloud "f composed with g of x."

Note: Composition is not commutative. That is, (f ° g)(x) is usually different from (g ° f)(x). The example below illustrates this.

Example: f(x) = 3x² + 12x – 1 and g(x) = 4x + 1

¹(f ° g)(x) = 3(4x + 1)² + 12(4x + 1) – 1

= 3(16x² + 8x + 1) + 48x + 12 – 1
= 48x² + 72x + 14

¹(g ° f)(x) = 4(3x² + 12x – 1) + 1

= 12x² + 48x – 4 + 1
= 12x² + 48x – 3