So upon solving some trigonometric equations, I found myself using the following method often:$$f[g(x)]=h(x)$$$$f[g(g^{-1}(x))]=h[g^{-1}(x)]$$$$f(x)=h[g^{-1}(x)]$$Which is how I usually find $f(x)$ when another function is inside it.
But is there a name for this method and/or anything I should note?
This is sometimes known as "precomposition by $g^{-1}(x).$" As long as the function of $g^{-1}$ exists, there's nothing you need to worry about.