Results 1 to 4 of 4

Math Help - Create a decomposition of functions

  1. #1
    Newbie
    Joined
    Oct 2012
    From
    new york
    Posts
    5

    Thumbs up Create a decomposition of functions

    Decomposing a function as a composition
    Find f(x) and g(x) such that h(x)=(fog)(x)

    h(x)=x2-3x/square root of x+5


    I tried to do it but I do not think is right. this is the answer I get: f(x)= x2-3x g(x)= 1/square root of x+5
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    Re: Create a decomposition of functions

    Your answer was a quotient, not a composition. You found f and g so that h(x) = f(x)/g(x).

    There are an unlimited number of choices given the way the problem is stated.
    For instance, there are always these trivial choices:
    Choice #1) f(x) = h(x) and g(x) = x
    Choice #2) f(x) = x and g(x) = h(x)
    Maybe more interesting is letting g(x) = x+5.

    Then h(x) = \frac{x^2-3x}{\sqrt{x+5}} and x = g(x)-5, so

    h(x) = \frac{(g(x)-5)^2-3(g(x)-5)}{\sqrt{(g(x)-5)+5}}

    = \frac{(g(x)^2 - 10g(x) + 25) + (-3g(x)+15))}{\sqrt{g(x)}}

    = \frac{g(x)^2 - 13g(x) + 40}{\sqrt{g(x)}}

    = f(g(x)), where f(t) = \frac{t^2 - 13t + 40}{\sqrt{t}}

    Choice #3) f(t) = \frac{t^2 - 13t + 40}{\sqrt{t}}, g(x) = x+5

    Now try g(x) = \sqrt{x+5}. Note that g(x) \ge 0 (actually, it can't be allowed to be 0). Have x = g(x)^2 - 5.

    Thus h(x) = \frac{x^2-3x}{\sqrt{x+5}} and x = g(x)^2-5, so

    Thus h(x) = \frac{(g(x)^2-5)^2-3(g(x)^2-5)}{\sqrt{(g(x)^2-5)+5}}

    = \frac{(g(x)^4-10g(x)^2+25) +(-3g(x)^2 + 15)}{\sqrt{g(x)^2}}

    = \frac{g(x)^4 -13 g(x)^2 + 40}{|g(x)|} (now use g(x) \ge 0 to know that |g(x)| = g(x))

    = \frac{g(x)^4 -13 g(x)^2 + 40}{g(x)}

    = g(x)^3 - 13g(x) + \frac{40}{g(x)}

    = f(g(x)), where f(t) = t^3 - 13t + \frac{40}{t}

    Choice #4) f(t) = t^3 - 13t + \frac{40}{t}, g(x) = \sqrt{x+5}
    Last edited by johnsomeone; October 8th 2012 at 04:31 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2012
    From
    new york
    Posts
    5

    Re: Create a decomposition of functions

    Hi! Thanks for answering my question. However my professor said that is not the way he wants it. He said that the answer g(x) should be g(x)=1/square root of x+5 and I only need to find f(x). I guess he only wants a quotient? Mine is wrong.Can you help me please?? So when I find (fog) I will end up with: x^2-3x/ square root of x+5 back again. Thanks!!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    Re: Create a decomposition of functions

    Review what I did for Choice #3 and Choice #4. In each case, I began with choice of g(x) (one time was x+5, the other it was sqrt(x+5)). Then I did some algebra and eventually found f(t) such that h(x) = f(g(x)). So if you repeat that process, you'll solve the problem. Repeat the process I did, only now your choice of g(x) will be:

    g(x)  = \frac{1}{\sqrt{x+5}}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. how to create a sequence s1 s2 s3 ..s11 s12....
    Posted in the Math Software Forum
    Replies: 6
    Last Post: June 4th 2010, 11:31 PM
  2. how to create def symbol?
    Posted in the LaTeX Help Forum
    Replies: 4
    Last Post: October 10th 2009, 12:42 AM
  3. Jordan Decomposition to Schur Decomposition
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 30th 2009, 01:52 PM
  4. create a polynomial...
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: August 30th 2008, 03:18 PM
  5. Symmetry "decomposition" of functions
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: June 15th 2008, 06:48 PM

Search Tags


/mathhelpforum @mathhelpforum