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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 $x$?

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  • $\begingroup$ Consider the Kolmogorov result that any real-valued continuous function on the unit interval can be represented by a Kolmogorov network. $\endgroup$ – Wuestenfux Feb 11 at 7:42
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    $\begingroup$ 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. $\endgroup$ – drhab Feb 11 at 7:50
  • $\begingroup$ 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? $\endgroup$ – marshal craft Feb 11 at 7:53
  • $\begingroup$ @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. $\endgroup$ – Kabir Shah Feb 11 at 7:55
  • $\begingroup$ @KabirShah Will you allow $g(x)=x$, $h(x)=x^2+1$, $k(x,y) = \frac{x}{y}$, so $k(g(x), h(x)) = \frac{x}{x^2+1}$ ? $\endgroup$ – gandalf61 Feb 11 at 8:56

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