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anonymous

  • one year ago

f(x)= x-7/x+2 g(x)=-2x-7/x-3 find f(g(x)) and g(f(x))

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  1. anonymous
    • one year ago
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    G(x)= -2x-7/x-1 (correction)

  2. anonymous
    • one year ago
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    prepare to do a raft of algebra

  3. anonymous
    • one year ago
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    Ok

  4. anonymous
    • one year ago
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    \[f(x)=\frac{x-7}{x+2}\\ g(x)=\frac{-2x-7}{x-1}\]

  5. anonymous
    • one year ago
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    that right?

  6. anonymous
    • one year ago
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    Yes :)

  7. anonymous
    • one year ago
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    ok lets to \[f(g(x))\] first

  8. anonymous
    • one year ago
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    what i am going to do is use cut and paste to put \[\frac{-2x-7}{x-1}\] everywhere i see an \(x\) in \[\frac{x-7}{x+2}\]

  9. anonymous
    • one year ago
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    \[\large f(g(x))=f(\frac{-2x-7}{x-1})=\frac{\frac{-2x-7}{x-1}-7}{\frac{-2x-7}{x-1}+2}\]

  10. anonymous
    • one year ago
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    clear what i did? that is the first step, bunch of algebra comes next

  11. anonymous
    • one year ago
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    Yes I understand so far :)

  12. anonymous
    • one year ago
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    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 \[f(g(x))=x\]

  13. anonymous
    • one year ago
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    Ok :D

  14. anonymous
    • one year ago
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    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)

  15. anonymous
    • one year ago
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    \[\frac{-2x-7-7(x-1)}{-2x-7+2(x-1)}\] is step one

  16. anonymous
    • one year ago
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    notice the judicious use of parentheses

  17. anonymous
    • one year ago
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    now remove the parentheses using the distributive property to get \[\frac{-2x-7-7x+7}{-2x-7+2x-2}\]

  18. anonymous
    • one year ago
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    combine like terms, and since they are inverses you will get an orgy of cancellation \[\frac{-9x}{-9}=x\]finished

  19. anonymous
    • one year ago
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    method for doing the second one is similar

  20. anonymous
    • one year ago
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    That makes sense

  21. anonymous
    • one year ago
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    I am a bit confused for the second one

  22. anonymous
    • one year ago
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    same as the first, only inside out

  23. anonymous
    • one year ago
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    I put..g(f(x))= -2(x-7/x+2)-7/(x-7/x+2)-1

  24. anonymous
    • one year ago
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    try it and see what happens it will be easy enough to know if you are right or not, because the answer will be \(x\)

  25. anonymous
    • one year ago
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    yeah that looks right

  26. anonymous
    • one year ago
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    However after this step I wasn't positive if I should multiply both by x+2 or distribute

  27. anonymous
    • one year ago
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    yes multiply top and bottom by \(x+2\) like i did before, carefully, using parentheses

  28. anonymous
    • one year ago
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    Ok I'll try that :)

  29. anonymous
    • one year ago
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    I don't think I'm doing this correctly ::

  30. anonymous
    • one year ago
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    :/

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