Composition and Inverse Functions#

Mathematics for Geoscientists

Injection, Surjection, Bijection#

Before we start talking about inverse functions, let us define some very important terms.

A function \(f: A \to B\) is:

  1. an injection (or injective or one-to-one) if different elements of the domain are assigned different elements of the codomain, i.e.:

\[ \forall a, a' \in A, (a \neq a') \Rightarrow (f(a) \neq f(a')). \]

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.

  1. 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,

\[ \forall b \in B, \exists a \in A : f(a) = b. \]

  1. a bijection (or bijective or one-to-one and onto or one-to-one correspondence) if it is both an injection and a surjection. That is,

\[ \forall b \in B, \exists ! a \in A: f(a) = b. \]

Let us summarise this in a table (source: Wikipedia):

surjective

non-surjective

injective

bijective

injective-only

non-injective

surjective-only

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 \(g: \mathbb{R} \to \mathbb{R}, g(x) = x^2\):

\[\begin{split} (g \circ f)(x) = (x+1)^2 \\ (f \circ g)(x) = x^2 + 1 \end{split}\]

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 \(h: \mathbb{R} \to \mathbb{R}, h(x) = \sqrt{x^2}\):

\[\begin{split} [h \circ (g \circ f)](x) = h[\sin(x + 1)] = \sqrt{\sin^2(x+1)} \\ [(h \circ g) \circ f](x) = \sqrt{ \sin^2[f(x)] } = \sqrt{\sin^2(x+1)} \end{split}\]

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 \(y \in B\):

\[ g[f(x)] = x \iff f[g(y)] = y. \]

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 \(f^{-1}\)