设 $f\left( x \right)$，$g\left( x \right)$，$h\left( x \right)$ 是 ${\mathbb{R}}$ 上的任意实值函数，如下定义两个函数 $\left(f \circ g\right)\left(x\right)$ 和 $\left( {f \cdot g} \right)\left( x \right)$；对任意 $x \in {\mathbb{R}}$，$\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)$；$\left( {f \cdot g} \right)\left( x \right) = f\left( x \right)g\left( x \right)$，则下列等式恒成立的是 \((\qquad)\)
【难度】
【出处】
2011年高考广东卷（文）
【标注】
【答案】
B
【解析】
对A选项：
$\left( {\left( {f \circ g} \right) \cdot h} \right)\left( x \right) = \left( {f \circ g} \right)\left( x \right)h\left( x \right) = f\left( {g\left( x \right)} \right)h\left( x \right)$，
$\left( {\left( {f \cdot h} \right) \circ \left( {g \cdot h} \right)} \right)\left( x \right) = \left( {f \cdot h} \right)\left( {\left( {g \cdot h} \right)\left( x \right)} \right) = \left( {f \cdot h} \right)\left( {g\left( x \right) \cdot h\left( x \right)} \right)= f\left( {g\left( x \right) \cdot h\left( x \right)} \right)h\left( {g\left( x \right) \cdot h\left( x \right)} \right)$；
故排除A．
对B选项：
$\left( {\left( {f \cdot g} \right) \circ h} \right)\left( x \right) = \left( {f \cdot g} \right)\left( {h\left( x \right)} \right) = f\left( {h\left( x \right)} \right)g\left( {h\left( x \right)} \right)$，
$\left( {\left( {f \circ h} \right) \cdot \left( {g \circ h} \right)} \right)\left( x \right) = \left( {f \circ h} \right)\left( x \right)\left( {g \circ h} \right)\left( x \right) = f\left( {h\left( x \right)} \right)g\left( {h\left( x \right)} \right)$；
故选B．
对C选项：
$\left( {\left( {f \circ g} \right) \circ h} \right)\left( x \right) = \left( {f \circ g} \right)\left( {h\left( x \right)} \right) = f\left( {g\left( {h\left( x \right)} \right)} \right)$，
$\left( {\left( {f \circ h} \right) \circ \left( {g \circ h} \right)} \right)\left( x \right) = \left( {f \circ h} \right)\left( {\left( {g \circ h} \right)\left( x \right)} \right) = \left( {f \circ h} \right)\left( {g\left( {h\left( x \right)} \right)} \right) = f\left( {h\left( {g\left( {h\left( x \right)} \right)} \right)} \right)$；
故排除C．
对D选项：
$\left( {\left( {f \cdot g} \right) \cdot h} \right)\left( x \right) = \left( {f \cdot g} \right)\left( x \right)h\left( x \right) = f\left( x \right)g\left( x \right)h\left( x \right)$，
$\left( {\left( {f \cdot h} \right) \cdot \left( {g \cdot h} \right)} \right)\left( x \right) = \left( {f \cdot h} \right)\left( x \right)\left( {g \cdot h} \right)\left( x \right) = f\left( x \right)h\left( x \right)g\left( x \right)h\left( x \right)$，
故排除D．
$\left( {\left( {f \circ g} \right) \cdot h} \right)\left( x \right) = \left( {f \circ g} \right)\left( x \right)h\left( x \right) = f\left( {g\left( x \right)} \right)h\left( x \right)$，
$\left( {\left( {f \cdot h} \right) \circ \left( {g \cdot h} \right)} \right)\left( x \right) = \left( {f \cdot h} \right)\left( {\left( {g \cdot h} \right)\left( x \right)} \right) = \left( {f \cdot h} \right)\left( {g\left( x \right) \cdot h\left( x \right)} \right)= f\left( {g\left( x \right) \cdot h\left( x \right)} \right)h\left( {g\left( x \right) \cdot h\left( x \right)} \right)$；
故排除A．
对B选项：
$\left( {\left( {f \cdot g} \right) \circ h} \right)\left( x \right) = \left( {f \cdot g} \right)\left( {h\left( x \right)} \right) = f\left( {h\left( x \right)} \right)g\left( {h\left( x \right)} \right)$，
$\left( {\left( {f \circ h} \right) \cdot \left( {g \circ h} \right)} \right)\left( x \right) = \left( {f \circ h} \right)\left( x \right)\left( {g \circ h} \right)\left( x \right) = f\left( {h\left( x \right)} \right)g\left( {h\left( x \right)} \right)$；
故选B．
对C选项：
$\left( {\left( {f \circ g} \right) \circ h} \right)\left( x \right) = \left( {f \circ g} \right)\left( {h\left( x \right)} \right) = f\left( {g\left( {h\left( x \right)} \right)} \right)$，
$\left( {\left( {f \circ h} \right) \circ \left( {g \circ h} \right)} \right)\left( x \right) = \left( {f \circ h} \right)\left( {\left( {g \circ h} \right)\left( x \right)} \right) = \left( {f \circ h} \right)\left( {g\left( {h\left( x \right)} \right)} \right) = f\left( {h\left( {g\left( {h\left( x \right)} \right)} \right)} \right)$；
故排除C．
对D选项：
$\left( {\left( {f \cdot g} \right) \cdot h} \right)\left( x \right) = \left( {f \cdot g} \right)\left( x \right)h\left( x \right) = f\left( x \right)g\left( x \right)h\left( x \right)$，
$\left( {\left( {f \cdot h} \right) \cdot \left( {g \cdot h} \right)} \right)\left( x \right) = \left( {f \cdot h} \right)\left( x \right)\left( {g \cdot h} \right)\left( x \right) = f\left( x \right)h\left( x \right)g\left( x \right)h\left( x \right)$，
故排除D．
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