AEF.CSS Composition: Socks and Shoes
The relation of composition in solving equations has been studied in several previous sections when one of the composing functions was linear.
This relation is a result of the important theorem for inverses of composite functions:
Theorem AEF.CSS:  If $f(x)= (f_2 \circ f_1)(x) =f_2(f_1(x)) = b$ then $f_1(x) =a \in f_2^{-1}(b)$ and  $x \in f_1^{-1}(a)$.
Thus all the solutions to the equation $f(x)=b$ correspond to all the solutions to  $f_1(x)=a$ where $f_2(a)=b$.
Thus the set $f^{-1}(b) =\{x : f(x)=b\} = f_1^{-1}(f_2^{-1}(b))$. ("socks and shoes").
If $ f: x \to b$, then $(f_2 \circ f_1): x \to f_1(x)=a \to b=f_2(a)$ and so $x \in f_1^{-1}(f_2^{-1}(b))$.

A mapping diagram can visualize this result by visually following arrows in the mapping diagrams back from $b$ to all $a$ where $f_2(a) = b$ and then back from such $a$ to all $x$ where $f_1(x)=a$.
Example AEF.CSS.1  Visualize the solutions of $f(x)=\sin^2(x)-4 =-3.36 $ for $x \in [-4,4]$.

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