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Functions and Rearranging Help

  1. Oct 15, 2014 #1
    Two functions are defined as:
    f(x) = 1/(-10x+14)+17
    g(x) = 1/(-4x-3)

    Find fg(x) and gf(x) and simplify answer into form (ax+b)/(cx+d) where a,b,c and d are numbers to be found.

    I know it is common practice to put in some of my own attempts onto here, but I am not even getting remotely close to an answer. :s

    Any help or guidance would be greatly appreciated!!
  2. jcsd
  3. Oct 15, 2014 #2
    fg(x) is defined to be equal to f(x)g(x) for all x in the intersection of the domains of f and g, so you just multiply the equations together.

    But it sounds like you are expecting fg and gf to be two different functions, which suggests there's a typo in your question.

    edit: there has to be a typo in your question, because fg and gf wouldn't be of the form (ax+b)/(cx+d)
  4. Oct 15, 2014 #3


    Staff: Mentor

    For the first function, what you wrote is
    $$ f(x) = \frac{1}{-10x + 14} + 17$$
    Is that what you intended to write?
  5. Oct 15, 2014 #4


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    Science Advisor
    Homework Helper

    I think the question is to evaluate the compositions, not the product. I.e. f(g(x)) and g(f(x)). Those are different and are of that form.
  6. Oct 15, 2014 #5


    Staff: Mentor

    That's my take as well, Dick.
  7. Oct 15, 2014 #6


    User Avatar

    Staff: Mentor

    At least show us how you started. Then we can tell you if you're starting off OK, and if not, give you hints on which way you should be going.
  8. Oct 15, 2014 #7


    Staff: Mentor

    It's not just common practice - it's a requirement of this board.
  9. Oct 15, 2014 #8
    From what I understood of the question and from the previous questions is to substitute g(x) into f(x), which I think you're right, it would be written like f(g(x)) and vice versa for g(f(x)). Sorry about not making this clearer, this is my first time learning about functions.

    I attempted to substitute 1/(-4x-3) into 1/(-10x+14)+17 to give 1/(-10(1/(-4x-3))+14)+17 (sorry I can't make these equations easier to read, I'm not sure how to).

    From there on I got a bit lost with the simplifying.
  10. Oct 15, 2014 #9
    Am I on the right track by substituting one equation into the other and and I just need to look back at my simplifying or am I going about this completely wrong?
  11. Oct 15, 2014 #10


    Staff: Mentor

    What you did in your previous post looks like you're on the right track.
  12. Oct 15, 2014 #11


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    Homework Helper

    You have 1/(-10(1/(-4x-3))+14)+17 = f(g(x)).
    To simplify, ##\frac{1}{-10 \left(\frac{1}{-4x-3} \right)+14}+17##, first, combine terms on the bottom of the fraction to get something of the form ##\frac{1}{\frac{ax+b}{cx+d}}+17##
    Next, note that ##\frac{1}{\frac{ax+b}{cx+d}}=\frac{cx+d}{ax+b}##
  13. Oct 15, 2014 #12

    You can also write ##f \circ g (x) = f(g(x)), g\circ f (x) = g(f(x))## if you're looking for notation.
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