1.3.5: Composition of Functions

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We saw in section 3.4 that the inverse \(f^{-1}\) of a bijection \(f\) is itself a function. Another operation on functions is composition: we can define a new function by composing two functions, \(f\) and \(g\), i.e., by first applying \(f\) and then \(g\). Of course, this is only possible if the ranges and domains match, i.e., the range of \(f\) must be a subset of the domain of \(g\). This operation on functions is the analogue of the operation of relative product on relations from section 2.7.

A diagram might help to explain the idea of composition. In Figure \(\PageIndex{1}\), we depict two functions \(f \colon A \to B\) and \(g \colon B \to C\) and their composition \((\comp{f}{g})\). The function \((\comp{f}{g}) \colon A \to C\) pairs each element of \(A\) with an element of \(C\). We specify which element of \(C\) an element of \(A\) is paired with as follows: given an input \(x \in A\), first apply the function \(f\) to \(x\), which will output some \(f(x) = y \in B\), then apply the function \(g\) to \(y\), which will output some \(g(f(x)) = g(y) = z \in C\).

Figure \(\PageIndex{1}\): The composition \(g \circ f\) of two functions \(f\) and \(g\).

Definition \(\PageIndex{1}\): Composition

Let \(f\colon A \to B\) and \(g\colon B \to C\) be functions. The composition of \(f\) with \(g\) is \(\comp{f}{g} \colon A \to C\), where \((\comp{f}{g})(x) = g(f(x))\).

Example \(\PageIndex{1}\)

Consider the functions \(f(x) = x + 1\), and \(g(x) = 2x\). Since \((\comp{f}{g})(x) = g(f(x))\), for each input \(x\) you must first take its successor, then multiply the result by two. So their composition is given by \((\comp{f}{g})(x) = 2(x+1)\).

Problem \(\PageIndex{1}\)

Show that if \(f \colon A \to B\) and \(g \colon B \to C\) are both injective, then \(\comp{f}{g}\colon A \to C\) is injective.

Problem \(\PageIndex{2}\)

Show that if \(f \colon A \to B\) and \(g \colon B \to C\) are both surjective, then \(\comp{f}{g}\colon A \to C\) is surjective.

Problem \(\PageIndex{3}\)

Suppose \(f \colon A \to B\) and \(g \colon B \to C\). Show that the graph of \(\comp{f}{g}\) is \(R_f \mid R_g\).