I want to find a function $$h = g \circ f$$ such that condition number of $g$ and $f$ are greater than $10$, but the condition number of $h$ is less than $1$.
I am trying to use polynomials like $x^{10}$ for $f$ and $g$, but I find that the condition number of $h$ can't be less than $1$. If I fix the condition number of $h$ to be less than $1$ and the condition number of $f$ to be greater than $10$, then I find that the condition number of $g$ must be less than $1$.
Typically, the condition number of a composition is the product of the condition numbers. Specifically, if $x \not =0$ and $h(x) \not = 0$, then the relative condition number of the function $h$ at the point $x$ is given by $$ \kappa_h(x) = \left | \frac{xh'(x)}{h(x)} \right|.$$ Now if $h = f \circ g$ and the chain rule can be applied, then $$ \frac{xh'(x)}{h(x)} = \frac{x f(g(x))g'(x)}{f(g(x))} = \frac{y f'(y)}{f(y)}\frac{x g'(x)}{g(x)}$$ where $y = g(x)$ and we assume that $y \not = 0$. It follows that $$\kappa_h(x) = \kappa_f(g(x)) \kappa_g(x).$$ In particular, if $\kappa_f(g(x)) \ge 10$ and $\kappa_g(x) \ge 10$, then $\kappa_h(x) \ge 100$ is inevitable.