E-CO04 · Recover a rule, then change its domain
Initially \(f(x)=x^2+ax+b\) and \(g(x)=x+2\) are defined on \(\mathbb R\). It is given that \(fg(x)-gf(x)=4x+10\) for every real \(x\), and \(f(-1)=2\).
Find \(a\) and \(b\).
Now restrict \(f\) to \(x\ge-2\), while keeping \(g\) on \(\mathbb R\). Find the largest restriction of \(D_g\) that permits \(fg\), and give the rule and range of that restricted composite.
Using the NEW domains in (b), solve \(fg(x)=gf(x)\).
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