例题 5. 5 设 f,g均为实数集上的函数,且 f( x) = 2x + 1,g( x) = x2 + 1,那么:
f  g( x) = f( g( x) ) = 2( x2 + 1) + 1 = 2x2 + 3,
g  f( x) = g( f( x) ) = ( 2x + 1)
+ 1 = 4x2 + 4x + 2,
f  f( x) = f( f( x) ) = 2( 2x + 1) + 1 = 4x + 3,
g  g( x) = g( g( x) ) = ( x2 + 1)
+ 1 = x4 + 2x2 + 2.
同关系的合成运算一样,函数的合成运算不满足交换律,但满足结合律,即:
( f  g