algebra question, Pexam PCAT #21, pg 58

[tivon]

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f(x)=3x +2
(gof)(x)=6x +7
f^-1 (x) =(x-2)/3

what is the value of g(x)?

The answer is g(x)= (gof)(f^-1 (x))= 2x+3.

I don't understand anything about this question and the answer. What is (gof)(x)? How did they get the formula of g(x) in the answer?Thanks.

 

[ForcedEntry]

if f(x) = 3x+2
and g(x) = 2x+3
then (gof)(x) = 2(f(x))+3 = 2(3x+2)+3 = 6x+7

therefore, to work this problem backwards:
if (gof)(x) = 6x+7
and f^-1(x) = (x-2)/3
then g(x) = 6[f^-1(x)]+7 = 6[(x-2)/3]+7 = 2(x-2)+7 = 2x+3

This is a good source of info: http://www.sosmath.com/

 

[acetyl]

What forced entry said is correct, but was kind of hard for me to follow. This is how I solved the question.

g(f(x))= 6x+7, then g(f(f^-1(x))) = g(x)

then, g(f(f^-1(x)))= 6((x-2)/3)+7
= 6x/3 - (6*2)/3 +7
= 2x -4 +7
= 2x+3

 

[tivon]

Thanks guys,
I took calculus, but didn't take algebra and precalculus in college. Is it true that:
f(f^-1(x))=x ?????
fog(x) = f(g(x)) ????

Thanks in advance.

 

[acetyl]

I couldn't find the concrete mathematical relationship earlier, but I know that:

f(x)=y, then f^-1👍=x