Given any function $f(x)$, how can you come up with the corresponding $g(x)$ such that $g(p)=p$ (where p is the root)?
Say, $$f(x)= sinx -\frac{x}{1.4}$$ my professor told me to simply isolate for $x$ and get: $$g(x)=1.4sinx$$
Why does this work and why can't I come up with any random function $g(x)$?
Thanks.
A fixed point is a point that satisfies $h(x) = x$. Note, I use $h$ here because I don't want to conflate the nomenclature with your $f$ and $g$.
Now, suppose we set up a fixed point approach for $g(x)\stackrel{\textrm{def}}{=} 1.4\sin x$:
$$g(x) = x \iff 1.4\sin x = x \iff \sin x = \frac{x}{1.4} \iff \sin x - \frac{x}{1.4} = 0.$$
If you start from the right, you can follow the arrows backwards to obtain $g(x)$.
If $x$ is the root, then $\sin x - \dfrac{x}{1.4} = 0$. So $x = 1.4 \sin x$.
If $f(p)=0$ then the following function with additive constant, $f+p$ will have $p$ as a fixed point. And so is the function $pe^{cf(x)}$ for an arbitrary non-zero constant $c$. So 'associated function' $g$ is not really a well-defined concept.
