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Let \(f(x)=3x+2\) and \(g(x)=ax+b\) , for some constants \(a\)and \(b\). If \(ab=20\) and \(f(g(x))=g(f(x))\) for \(x=0,1,2\ldots 9,\) fin the sum of all possible values of \(a\).

 Dec 5, 2018
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\(f(g(x) = g(f(x))\\ f(g(x) = 3(ax+b)+2 = 3ax + (3b+2)\\ g(f(x)) = a(3x+2)+b = 3ax + (2a+b)\\ 3b+2 = 2a+b\\ 2b+2=2a\\ b=a-1\)

 

\(ab=20\\ a(a-1)=20\\ a^2-a-20=0\\ (a-5)(a+4)=0\\ a=5 \vee a=-4\\ 5+(-4)=1\)

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 Dec 5, 2018

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