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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...9, find the sum of all possible values of a.

 Aug 2, 2021
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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 $x=0,1,2...9$ find the sum of all possible values of $a$.

 

For nested functions, we take the interior function and substitute it into $x$ of the outer function. 

 

$f(g(x)) = 3ax + 3b + 2$

$g(f(x)) = 3ax + 2a + b$

 

$3ax + 3b + 2 = 3ax + 2a + b \rightarrow 2a = 2b+2$

 

Thus, we need to solve the system of equations 

$ab = 20$

$2a = 2b + 2$

 

We can either solve for a variable and substitute into the other equation, or we can guess and check to see that our solutions are 

$a = 5, b = 4$

$a = -4, b= -5$

 

Thus the sum of all possible values of $a$ is $5 - 4 = \boxed{1}$. 

 Aug 2, 2021

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