Composition and Inverse Functions
Contents
Composition and Inverse Functions#
Injection, Surjection, Bijection#
Before we start talking about inverse functions, let us define some very important terms.
A function \(f: A \to B\) is:
an injection (or injective or onetoone) if different elements of the domain are assigned different elements of the codomain, i.e.:
We can determine easily whether a function is an injection from its graph. For a function to be injective, any horizontal line must cross its graph in at most one point. This is why the introduced notion of strictly monotonic function is important: if a function is strictly monotonic (on some interval or on its entire domain), then it is an injection.
a surjection (or surjective or onto) if every element of its codomain \(B\) is a possible value of \(f(a)\) for some \(a \in A\). That is, the image of \(f\) must be equal to its codomain. That is,
a bijection (or bijective or onetoone and onto or onetoone correspondence) if it is both an injection and a surjection. That is,
Let us summarise this in a table (source: Wikipedia):
surjective 
nonsurjective 


injective 
bijective 
injectiveonly 
noninjective 
surjectiveonly 
general 
Function composition#
Let \(f: A \to B\) and \(g: C \to D\) be two functions. If \(f[A] \subseteq C\), we can define a function \(h: A \to D\) such that \(h(x) = g[f(x)], \forall x \in A\). We denote the function \(h\) defined this way by \(g \circ f\) and call it the composition of \(g\) and \(f\).
This is generally not a commutative operation. For \(f: \mathbb{R} \to \mathbb{R}, f(x) = x + 1\) and \(g: \mathbb{R} \to \mathbb{R}, g(x) = x^2\):
But it is associative. For \(f: \mathbb{R} \to \mathbb{R}, f(x) = x + 1, g: \mathbb{R} \to \mathbb{R}, g(x) = \sin(x)\) and \(h: \mathbb{R} \to \mathbb{R}, h(x) = \sqrt{x^2}\):
As an example of a function composition that is not defined, consider \(f: \mathbb{R} \to \mathbb{R}\) where \(f(x) = \cos x\) and \(g: \mathbb{R} \to \mathbb{R}\) where \(g(x) = \sqrt{x}\). The range of \(f\) is \([1, 1]\) but the natural domain of \(g\) is \([0, + \infty)\).
Inverse function#
Let \(f: A \to B\) be a function and assume there exists another function \(g: B \to A\) such that for all \(x \in A\) and for all \(y \in B\):
We then call \(g\) the inverse function of \(f\) and we denote it \(g = f^{1}\).
Properties
A function \(f\) is invertible iff it is a bijection
The inverse function \(f^{1}\) is unique.
The graph of the inverse function \(G(f^{1})\) is symmetric to the graph \(G(f)\) with respect to \(y = x\).
Monotonic properties of \(f\) are preserved on \(f^{1}\)