How to define functions without using variables

by Jakub Marian

Tip: See my list of the Most Common Mistakes in English. It will teach you how to avoid mis­takes with com­mas, pre­pos­i­tions, ir­reg­u­lar verbs, and much more (PDF Version).

First, note that this article is not about the definition of elementary functions, such as $\sin$ and $\cos$, and operations, like $a^b$, but rather about the way to define a new function using elementary functions, e.g. $f(x) = x^2e^x$, without the need to use variables.

There already is a way to denote functions without variables for many of them; for example, if $A$ is an operator which can be applied to a real function, you can write just $A(\sin)$ instead of $A(f)$, $f(x) = \sin(x)$. The reason why we generally don’t do that, i.e. why you are very unlikely to meet “$\sin$” without an argument (at least in the mathematical world) is that it would lead to an inconsistency in notation. For example, if $f(x) = \sin(x^2)$, how would you denote $A(f)$ without defining $f$ first using variables?

The solution is actually very simpledefine a new function:

$$ \id(x) = x\,. $$

Here I intentionally don’t write to which set the $x$ belongs. It can be $\id:ℝ→ℝ$, $\id:ℂ→ℂ$ or any other you find appropriate. I’ve chosen the symbol $\id$, which is typeset as {\large\iota} in Latex because it is the identity function, i.e. the function that “does nothing”, and it is already common to denote the inclusion from a set to its superset (i.e. the function that “does nothing”) by iota.

Using $\id$, we can denote any function without using variables. We will follow two rules:

  1. If $*$ is a binary operation and $f$ and $g$ are functions, then $f*g$ is a function defined by $$(f*g)(x) = f(x)*g(x)\,.$$ In particular, $fg$ means $(fg)(x) = f(x)g(x)$ and $\l(÷{f}{g}\r)(x) = ÷{f(x)}{g(x)}$. Also, for the operation of exponentiation, we let $f^g$ denote the function $$(f^g)(x) = (f(x))^{g(x)}\,.$$
  2. If $f$ and $g$ are functions, then the composition of $f$ and $g$, i.e. $f ∘ g$, may be denoted $f[g]$.

For example, using the first rule:

$$ \text{If}\quad\begin{array}[l] \phantom{} f = \id^3 \\ g = \sin^2 \\ h = \sin\cos \\ i = e^\id \\ j = ÷{\sin}{\cos} \end{array}\quad\text{then}\quad\begin{array}[l] \phantom{} f(x) = x^3 \\ g(x) = (\sin(x))^2 = \sin^2(x) \\ h(x) = \sin(x)\cos(x) \\ i(x) = e^x \\ j(x) = ÷{\sin(x)}{\cos(x)} \end{array} $$

And using the second rule:

$$ \text{If}\quad\begin{array}[l] \phantom{} f = \sin[\cos] \\ g = \tan[\id^2+1] \\ h = \log\l[÷{\sin}{\cos}\r]\end{array}\quad\text{then}\quad\begin{array}[l] \phantom{} f(x) = \sin(\cos(x)) \\ g(x) = \tan(x^2+1) \\ h(x) = \log\l(\l(÷{\sin}{\cos}\r)(x)\r) = \log\l(÷{\sin(x)}{\cos(x)}\r)\end{array} $$

Of course, the second rule is optional because we can always write $f ∘ g$ instead of $f[g],$ but using brackets is often more naturalwhich of the following expressions do you consider more legible?

$$ \sin[\id^2]\cos[\id^3] = ({\sin} ∘ \id^2)({\cos} ∘ \id^3) $$

Using square brackets instead of round parentheses allows us to distinguish between multiplication and composition. Compare

$$ (\cos+\sin)(\cos-\sin) = \cos^2-\sin^2 $$

with

$$ ({\cos}+{\sin})[{\cos}-{\sin}] = \cos[{\cos}-{\sin}] + \sin[{\cos}-{\sin}]\,, $$

What is this all good for?

Maybe you are wondering why we should even bother to define functions without using variables. Consider the following function:

$$ f = (\id^2+\id^3+\id^4)∘(2\id^2+3\id^3+4\id^4)∘(\id+\id^2) $$

Its structure is clear. However, in the standard notation, there’s no way to write $f$ using a simple formula. Subsequent polynomials are simply put “inside” the preceding ones:

\begin{align*} f(x) = &\big(2(x+x^2)^2+3(x+x^2)^3+4(x+x^2)^4\big)^2+\\ &+\big(2(x+x^2)^2+3(x+x^2)^3+4(x+x^2)^4\big)^3+\\ &+\big(2(x+x^2)^2+3(x+x^2)^3+4(x+x^2)^4\big)^4\,, \end{align*}

which is just a mess. In the former notation, it wouldn’t be hard to compute the value of $f$ at some point using only a pen and paper, for example

$$ f(1) =(\id^2+\id^3+\id^4) \l((2\id^2+3\id^3+4\id^4)(2)\r) = (\id^2+\id^3+\id^4)(96) \approx 86\cdot10^6 \ , $$

but it would be a dull task in the standard notation. And not only is the task dull, it is also much more demanding on computational resources when performed using a computer.

The simplification does not lie only in computation of numerical values, but also in symbolic manipulations. For example, using the chain rule for derivatives

$$ (f∘g∘h)’ = (f’∘g∘h)(g’∘h)h’ $$

we can compute

\begin{align*} f’ = &\l((2\id+3\id^2+4\id^3)∘(2\id^2+3\id^3+4\id^4)∘(\id+\id^2)\r)\\ &\l((4\id+9\id^2+16\id^3)∘(\id+\id^2)\r)(1+2\id)\,. \end{align*}

Can you imagine calculating the derivative in the traditional notation?

By the way, I have written several educational ebooks. If you get a copy, you can learn new things and support this website at the same time—why don’t you check them out?

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