In a mathematical calculation workflow where the output of an inner function $g(x)$ is used as the direct input for an outer function $f$, forming the composite function $(f \circ g)(x)$, what fundamental requirement must the value of $g(x)$ satisfy?

Input Compatibility in Multi-Stage Calculations

In a corporate financial model, an analyst uses a two-stage process where the output of function $g(x)$ is passed as the input to function $f$. For the multi-stage model $(f \circ g)(x)$ to be mathematically defined, the numerical value produced by $g(x)$ must be a valid element within the __________ of the function $f$.

In a professional data processing workflow modeled by the function composition $(f \circ g)(x)$, match each mathematical component with its specific requirement or role regarding domain compatibility.

In a professional logistics system, an analyst uses a two-stage mathematical model $(f \circ g)(x)$ to determine fuel surcharges. Arrange the following steps in the correct order to ensure the calculation is valid according to the domain requirement.