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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 \(x=0,1,2\ldots 9\), find the sum of all possible values of a.

 Oct 8, 2019
 #1
avatar+111438 
+2

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

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

 

Since these are equal then

 

3ax + 3b + 2  =  3ax + 2a + b

 

3b + 2  =  2a + b

 

2b + 2  =  2a

 

b + 1  =  a

 

b = a - 1

 

And since ab  = 20.....then

 

a (a - 1)  = 20

 

a^2  - a  = 20

 

a^2 - a - 20  = 0

 

(a - 5) (a + 4)  = 0

 

Setting each factor to 0  and solving for a produces  a = 5  or  a  = -4

 

So....the sum of these possible values for a   =  1

 

 

cool cool cool

 Oct 8, 2019
 #2
avatar+1200 
0

Thanks CPhill!!!

 Oct 8, 2019

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