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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

 

See also

Identity of an operation, identity function, inverse, composite

 


  this page updated 21-feb-16
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes