Neither full-domain composition works
8 written marks
Let \(f(x)=\sqrt{x-1}\), \(x\ge1\), and \(g(x)=3x+2\), \(-1\le x\le2\).
Part a · 2 marks
Decide whether \(fg\) can use all of \(D_g\), and whether \(gf\) can use all of \(D_f\). Justify both decisions.
Part b · 4 marks
In each order, restrict the inner domain as little as possible so that the composition is defined. Give each resulting rule and domain.
Part c · 2 marks
On the restricted domains in (b), solve \(fg(x)=2\) and, separately, \(gf(x)=5\).
Complete-answer checking, worked solutions and indicative written criteria.
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