# 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 $$f^{-1}$$