2.7.5 Function Composition

Understanding how to combine functions through composition (f∘g) and analyzing the domain and range of composite functions.

定义

Function composition is the process of combining two or more functions to create a new function. Given two functions \(f\) and \(g\), the composition of \(f\) with \(g\), denoted as \((f \circ g)(x)\) or \(f(g(x))\), is defined as applying function \(g\) first to the input \(x\), and then applying function \(f\) to the result. Formally, \((f \circ g)(x) = f(g(x))\). The domain of \(f \circ g\) consists of all values \(x\) in the domain of \(g\) such that \(g(x)\) is in the domain of \(f\). Function composition is not commutative, meaning \((f \circ g)(x) \neq (g \circ f)(x)\) in general. Composition is associative: \((f \circ (g \circ h))(x) = ((f \circ g) \circ h)(x)\).

核心公式

• \((f \circ g)(x) = f(g(x))\)
• \(\text{Domain of } (f \circ g) = \{x \in \text{Domain}(g) : g(x) \in \text{Domain}(f)\}\)
• \(\text{Range of } (f \circ g) \subseteq \text{Range}(f)\)
• \((f \circ g)(x) \neq (g \circ f)(x) \text{ (composition is not commutative)}\)
• \((f \circ (g \circ h))(x) = ((f \circ g) \circ h)(x) \text{ (composition is associative)}\)

易错点

• ⚠️ Reversing the order of composition: Students often confuse \((f \circ g)(x)\) with \((g \circ f)(x)\), forgetting that \(f \circ g\) means 'apply \(g\) first, then \(f\)', not the other way around.
• ⚠️ Ignoring domain restrictions: Students frequently fail to check whether the output of the inner function falls within the domain of the outer function, leading to incorrect domain specifications for the composite function.
• ⚠️ Assuming commutativity: Students mistakenly believe that \((f \circ g)(x) = (g \circ f)(x)\) always holds, when in fact function composition is not commutative and these typically produce different results.
• ⚠️ Incorrectly simplifying composite functions: When simplifying \((f \circ g)(x)\), students may make algebraic errors by not properly substituting the entire expression \(g(x)\) into \(f\), especially when \(g(x)\) is a complex expression.