The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.
Composite functions and their domains
Use fg(x)=f(g(x)) consistently.
Distinguish full-domain composition from an explicitly restricted composition.
Retain intermediate restrictions after simplification.
We use \(fg(x)=f(g(x))\): apply \(g\) first. To form \(fg\) on all of \(D_g\), require \(R_g\subseteq D_f\). If this fails, a question may explicitly ask you to restrict the inner domain to \(\{x\in D_g:g(x)\in D_f\}\). Do not silently change a stated domain.