anonymous
  • anonymous
f(x)= x-7/x+2 g(x)=-2x-7/x-3 find f(g(x)) and g(f(x))
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
G(x)= -2x-7/x-1 (correction)
anonymous
  • anonymous
prepare to do a raft of algebra
anonymous
  • anonymous
Ok

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anonymous
  • anonymous
\[f(x)=\frac{x-7}{x+2}\\ g(x)=\frac{-2x-7}{x-1}\]
anonymous
  • anonymous
that right?
anonymous
  • anonymous
Yes :)
anonymous
  • anonymous
ok lets to \[f(g(x))\] first
anonymous
  • anonymous
what i am going to do is use cut and paste to put \[\frac{-2x-7}{x-1}\] everywhere i see an \(x\) in \[\frac{x-7}{x+2}\]
anonymous
  • anonymous
\[\large f(g(x))=f(\frac{-2x-7}{x-1})=\frac{\frac{-2x-7}{x-1}-7}{\frac{-2x-7}{x-1}+2}\]
anonymous
  • anonymous
clear what i did? that is the first step, bunch of algebra comes next
anonymous
  • anonymous
Yes I understand so far :)
anonymous
  • anonymous
ok now before we do the algebra, i want to let you know that a miracle will occur the reason you were given this problem is that \(f\) is the inverse of \(g\) and vice versa that means when we get done we will find \[f(g(x))=x\]
anonymous
  • anonymous
Ok :D
anonymous
  • anonymous
staring here \[\frac{\frac{-2x-7}{x-1}-7}{\frac{-2x-7}{x-1}+2}\] get rid of the compound fraction by multiplying top and bottom by \(x-1\) (carefully)
anonymous
  • anonymous
\[\frac{-2x-7-7(x-1)}{-2x-7+2(x-1)}\] is step one
anonymous
  • anonymous
notice the judicious use of parentheses
anonymous
  • anonymous
now remove the parentheses using the distributive property to get \[\frac{-2x-7-7x+7}{-2x-7+2x-2}\]
anonymous
  • anonymous
combine like terms, and since they are inverses you will get an orgy of cancellation \[\frac{-9x}{-9}=x\]finished
anonymous
  • anonymous
method for doing the second one is similar
anonymous
  • anonymous
That makes sense
anonymous
  • anonymous
I am a bit confused for the second one
anonymous
  • anonymous
same as the first, only inside out
anonymous
  • anonymous
I put..g(f(x))= -2(x-7/x+2)-7/(x-7/x+2)-1
anonymous
  • anonymous
try it and see what happens it will be easy enough to know if you are right or not, because the answer will be \(x\)
anonymous
  • anonymous
yeah that looks right
anonymous
  • anonymous
However after this step I wasn't positive if I should multiply both by x+2 or distribute
anonymous
  • anonymous
yes multiply top and bottom by \(x+2\) like i did before, carefully, using parentheses
anonymous
  • anonymous
Ok I'll try that :)
anonymous
  • anonymous
I don't think I'm doing this correctly ::
anonymous
  • anonymous
:/

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