4

Can any function be written as a composition of other functions? For example, can a polynomial $h(x)$ be written as $k(g(x))$?

Farman H
  • 103
  • 4
    Well, any function $f(x)$ can be written as $f(g(x))$, where $g(x)=x$. – André Nicolas Dec 15 '15 at 06:26
  • Consider the case of $h(x)=x$ and $h(x)=k(g(x))$. Then $k$ is the functional inverse of $g$. When $g(x)=x^2$, that would imply that $k(x)=x^{\frac{1}{2}}$ which is not a polynomial. – JMoravitz Dec 15 '15 at 06:27

2 Answers2

2

This is called the "extension problem" or the "lifting problem" depending on which one you fix ($f$ or $g$, where $h = g \circ f$). Its answer depends on what field or ring your polynomials are over (or more generally, what category you are working in). For a nice introductory treatment see Lawvere and Schanuel's "Conceptual Mathematics" page 45.

JustAskin
  • 770
  • 5
  • 14
0

Every function can be written as a composition, using the identity function: $h = h \circ \text{id} = \text{id} \circ h$

Your particular example will only work for some polynomials. E.g. $h(x) = 2x - 42$ can not be composed using your given functions. $$ h = k \circ g \Rightarrow \\ k = h \circ g^{-1} $$ Thus $k(x) = 2\sqrt{x} - 42 $ which is not a polynomial.

mvw
  • 34,562