# Can any function be rewritten as a composition of functions on $x$?

Many functions use forms that have multiple instances of $$x$$, including polynomials and rational functions. However, finding the domain and range can be simpler when they are written with only one use of $$x$$. It can also make transformations more clear.

For example, the polynomial function $$f(x) = x^2-10x+35$$ can be rewritten as $$f(x) = \left(x-5\right)^2+10$$.

By composition of functions, I mean that:

$$f(x) = x^2-10x+35$$

But it can be rewritten as a composition of functions on $$x$$, with each composed function only using $$x$$ once:

$$f(x)=g(h(i(x)))$$ $$g(x)=x+10$$ $$h(x)=x^2$$ $$i(x)=x-5$$

However, I cannot figure out how some rational functions can be rewritten in this form, such as:

$$f(x)=\frac{x}{x^2+1}$$

Can any function be written with only one use of $$x$$?

• Consider the Kolmogorov result that any real-valued continuous function on the unit interval can be represented by a Kolmogorov network. – Wuestenfux Feb 11 at 7:42
• So what you call a "function on $x$" is a function "with only one use of $x$"? Then what if we write $h(x)=x\cdot x$ instead of $h(x)=x^2$? Before sensible things can be said first the vague concepts around must be defined rigorously. – drhab Feb 11 at 7:50
• I think not, and is related to if a number is algebraic, transcendental, and fundamental to notion of field extensions. See field extension.you seem to ask about field of integers, and it rational extension? – marshal craft Feb 11 at 7:53
• @drhab Sorry, I'll make it more clear that each composed function must have one use of $x$, I'm not sure if there is a more formal term for it. – Kabir Shah Feb 11 at 7:55
• @KabirShah Will you allow $g(x)=x$, $h(x)=x^2+1$, $k(x,y) = \frac{x}{y}$, so $k(g(x), h(x)) = \frac{x}{x^2+1}$ ? – gandalf61 Feb 11 at 8:56