How do I decompose a function when I'm given $f(g(x))$ and $f(x)$ and the required is $g(x)$? I done some searching on Google and most sites demonstrate the solution where it's left open, they just give $f(g(x))$ and let you solve to whatever set of functions you get (because there are multiple sets of functions that satisfy the decomposition).
Example: $(g \circ f)(x)= 3/x-2$
$f(x)= (x+1)/(x-2)$
$g(x)= ?$
1 Answer
If you let $h(x)$ be whatever expression you have on the right side, we have $f(g(x)) = (f \circ g)(x) = h(x)$, i.e., $f \circ g = h $. If $f$ is invertible, then we can write $g = f^{-1} \circ h$. Since $f$ is given, you can compute $f^{-1}$ by whatever means you already know.
But I think you meant examples like:
If $f(g(x)) = \sin{2x}$, and $f(x) = \sin{x}$, then $g(x) = 2x$, and so on. Always try to recognize operations like squaring, applying $\cos$ to something, taking logarithims, etc.
