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) = 3x^{2} + 12x – 1 and g(x) = 4x + 1


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


= 3(16x^{2} + 8x + 1) + 48x + 12 – 1
= 48x^{2} + 72x + 14


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


= 12x^{2} + 48x – 4 + 1
= 12x^{2} + 48x – 3 
See
also
Identity
of an operation, identity
function,
inverse, composite
