SOLUTION: let f(x)=3x-2x and g(f(x))=2x^2-3x+5. what is g(x)?

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Question 275302: let f(x)=3x-2x and g(f(x))=2x^2-3x+5. what is g(x)?
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
let f(x)=3x-2x and g(f(x))=2x^2-3x+5. what is g(x)?

I think you botched what %22f%28x%29%22 equals because 3x-2x is just x,
and if %22f%28x%29%22 were merely x, then %22g%28x%29%22 would just be the same as
%22g%28f%28x%29%29%22 which would be 2x%5E2-3x%2B5, and that would be a silly
problem.


So I will make up something else for %22f%28x%29%22, since I don't believe
you really meant %22f%28x%29%22=3x-2x. Suppose the problem were

let %22f%28x%29%22=x-2 and %22g%28f%28x%29%29%22=2x%5E2-3x%2B5. what is %22g%28x%29%22?

Then %22g%28x%29%22 would have to be a quadratic function, say 

%22g%28x%29%22=Ax%5E2%2BBx%2BC

Then 

%22g%28f%28x%29%29%22=A%28x-2%29%5E2%2BB%28x-2%29%2BC

%22g%28f%28x%29%29%22=A%28x%5E2-4x%2B4%29%2BBx-2B%2BC

%22g%28f%28x%29%29%22=Ax%5E2-4Ax%2B4A%2BBx-2B%2BC

Rearrange the terms:

%22g%28f%28x%29%29%22=Ax%5E2%2BBx-4Ax%2B4A-2B%2BC

Factor x out of the 2nd and 3rd terms on the right

%22g%28f%28x%29%29%22=Ax%5E2%2B%28B-4A%29x%2B4A-2B%2BC

Since %22g%28f%28x%29%29%22=2x%5E2-3x%2B5, we equate coefficients:

system%28A=2%2CB-4A=-3%2C4A-2B%2BC=5%29

Substitute A=2 into B-4A=-3

B-4%282%29=-3
B-8=-3
B=5

Substitute A=2 and B=5 into

4A-2B%2BC=5%29

4%282%29-2%285%29%2BC=5%29

8-10%2BC=5

-2%2BC=5

C=7

So %22g%28x%29%22=Ax%5E2%2BBx%2BC

becomes

 g%28x%29=2x%5E2%2B5x%2B7

Edwin