The function \(g\) is defined by \(g(x)=|x-1|-2 \). If \(g(a)=g(0)\) and \(a>0\), what is the value of \(a\)?

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Find \(g(a)\) and \(g(0)\).

$$ g(x)=|x-1|-2 $$ $$ g(a)=|a-1|-2 $$
$$ g(0)=|0-1|-2 $$ $$ = |-1|-2 $$ $$ = 1-2 $$ $$ = -1 $$

Setting \(g(a)=g(0) \):

$$|a-1|-2=-1$$ $$ |a-1|=1 $$ $$ a-1 = 1 \text{ or } -1 $$ $$ a= 2 \text{ or } 0 $$

Since \(a>0, a=\boxed{2} \)