What Is a Function?

  • All things are subject to interpretation whichever interpretation prevails at a given time is a function of power and not truth. (Friedrich Nietzsche)
  • The function of muscle is to pull and not to push, except in the case of the genitals and the tongue.(Leonardo da Vinci)
  • The supreme function of reason is to show man that some things are beyond reason. (Blaise Pascal)
  • The function of education is to teach one to think intensively and to think critically…Intelligence plus character – that is the goal of true education. (Martin Luther King, Jr.)

Functions f: \mathcal{Q}^m\rightarrow \mathcal{Q}^n

Definition of function according to Nasa

Domain

We denote by \mathcal{Q} the set of rational numbers, that is, the numbers with finite or periodic decimal expansion. We write f: D(f)\rightarrow \mathcal{Q} if for each given value of x in the domain D(f) of f, a rational number f(x) is assigned, that is, for each x\in D(f) the function value f(x)\in \mathcal{Q} is assigned.

Range

The set of values f(x) for x\in D(f) forms the range R(f) of f. We can thus write f:D(f)\rightarrow R(f) stating that for each x\in D(f) there is assigned a value f(x)\in R(f), and for each value y\in R(f) there is at least one value x\in D(f) such that y=f(x).

f: From Domain to Range

We write f:D(f)\rightarrow \mathcal{Q} to indicate that for each x\in D(f) a specific value f(x)\in \mathcal{Q} is assigned in the sense that x\rightarrow f(x).In particular, writing f:\mathcal{Q}\rightarrow \mathcal{Q}, means that for each x\in \mathcal{Q}, a function value f(x)\in \mathcal{Q} is assigned.

Writing f:\mathcal{Q}^m\rightarrow \mathcal{Q}^n means that for each m-vector x\in \mathcal{Q}^m, an n-vector f(x) with rational coefficients is assigned.

If f(x)\in \mathcal{Q}^n is defined for x in a subset \Omega of \mathcal{Q}^m, which we write \Omega\subset \mathcal{Q}^m expressing that if x\in\Omega then x\in \mathcal{Q}^m, then we write f:\Omega\rightarrow \mathcal{Q}^n.

We often write e.g. f: \mathcal{Q}^m\rightarrow \mathcal{Q}^n without explicitly specifying the domain D(f)\subset \mathcal{Q}^m, or the range R(f)\subset \mathcal{Q}^n.

It is common to denote by \mathcal{R} the set of all rational numbers together with the numbers with an infinite non-periodic decimal expansion, referred to as the set of \emph{real numbers}.

We shall see a function $f: \mathcal{Q}^m\rightarrow \mathcal{Q}^n$ can uniquely be extended to a function $f: \mathcal{R}^m\rightarrow \mathcal{R}^n$, if the function f(x) is continuous in a way to be specified precisely below (Lipschitz continuous).

We denote by \mathcal{R}_+ the set of positive real numbers.

Read More

  • \hyperref[whatisafunction]{What is a Function?}

To Think About

  • How did the notion of function develop?
  • Who introduced the concept and terminology

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