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Math Help - Combinations of Functions

  1. #1
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    Combinations of Functions

    I am having a fair bit of trouble with combinations of functions.

    First, I'd just like confirmation on whether or not I done a more basic question correctly.

    1.) If f(x)=x^2 -x -2, g(x)= x - 3 ,and  h(x) =2x determine the following:

    b.) f(h(g(0)))

    So
    g(0)=0-3
    g(0)=-3
    h(g(0))=2(-3)
    h(g(0))=-6
    f(h(g(0)))=f(-6)
    f(h(g(0)))=(-6)^2 -(-6) -2
    f(h(g(0)))=40

    Is this correct?

    For the next question I am given the following:



    I don't know how I am supposed to use the graph to evaluate each expression here. Also, is the expression for a.) the same thing as f(g(2))?

    Lastly I am asked:

    If f(x)=x^2 -x +2 and g(x)=x-2, find h(x) such that f(x)=g(h(x)).

    I have no idea how to solve this one.
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  2. #2
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    Re: Combinations of Functions

    Quote Originally Posted by BobRoss View Post
    For the next question I am given the following:
    The first is correct.
    BUT the second is based on the two graphs.
    a) g(2)=0 so from the graph f(0)=~?.
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  3. #3
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    Re: Combinations of Functions

    Sorry I'm still confused. Should I be looking at the x or y values on the graph?
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  4. #4
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    Re: Combinations of Functions

    You have to use that: (f o g)(x)=f(g(x))
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  5. #5
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    Re: Combinations of Functions

    Okay, so is the answer for a.) f(0)=4?
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    Re: Combinations of Functions

    Quote Originally Posted by BobRoss View Post
    Sorry I'm still confused. Should I be looking at the x or y values on the graph?
    Your looking for both.
    When x=2 then g(2)=0 so from the graph f(g(2))=f(0)=1.
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  7. #7
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    Re: Combinations of Functions

    Okay I think I see now. So for b,  f(2)=3 then g(3)=1

    And for c,  f(1)=2 then  f(2)=3

    Are these correct? Now what do I do for d?
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  8. #8
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    Re: Combinations of Functions

    Yes, those are correct. For (d) (f+ g)(2), use the definition, which I am sure you were given: f+ g is defined by (f+ g)(x)= f(x)+ g(x). From your graphs determine f(2) and g(2), then add those values.
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  9. #9
    MHF Contributor Siron's Avatar
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    Re: Combinations of Functions

    d) Note that (f+g)(2)=f(2)+g(2)=...

    @HallsofIvy:
    Sorry, I didn't see your post.
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  10. #10
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    Re: Combinations of Functions

    Okay so then since f(2)=3 and g(2)=0 then f(2)+g(2)=3+0=3 ?

    I have two more questions that I am stuck on. They are on the assignment but I can't find any examples of them in the text so I don't know how to approach them.

    1.) if f(x)=x^2 -x+2 and  g(x)=x-2, find h(x) such that  f(x) = g(h(x))

    2.) If g(x)=x-2 and  f(x)=3x-2, find x such that g(g(x))=f(f(x))
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  11. #11
    MHF Contributor Siron's Avatar
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    Re: Combinations of Functions

    1) What is g(h(x)) if you know that g(x)=x-2? Can you find h(x) now?
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    Re: Combinations of Functions

    Quote Originally Posted by BobRoss View Post
    Okay so then since f(2)=3 and g(2)=0 then f(2)+g(2)=3+0=3 ?
    1.) if f(x)=x^2 -x+2 and  g(x)=x-2, find h(x) such that  f(x) = g(h(x))

    2.) If g(x)=x-2 and  f(x)=3x-2, find x such that g(g(x))=f(f(x))
    I will do 1) if you will 2) and post the solution.
    h(x)=x^2-x+4
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  13. #13
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    Re: Combinations of Functions

    Gah, I still don't really know for either question. I just can't seem to wrap my head around combinations of functions, especially when working backwards through them. For the second question, what I've done so far is:

    g(x-2)=(x-2)-2
    g(x-2)=x-4

    f(3x-2)=3(3x-2)-2
    f(3x-2)=9x-6-2
    f(3x-2)=9x-8

    That doesn't seem to be correct though. What have I done wrong?
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  14. #14
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    Re: Combinations of Functions

    Quote Originally Posted by BobRoss View Post
    Gah, I still don't really know for either question. I just can't seem to wrap my head around combinations of functions, especially when working backwards through them. For the second question, what I've done so far is:

    g(x-2)=(x-2)-2
    g(x-2)=x-4

    f(3x-2)=3(3x-2)-2
    f(3x-2)=9x-6-2
    f(3x-2)=9x-8

    That doesn't seem to be correct though. What have I done wrong?
    Set them equal to one another and solve.
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  15. #15
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    Re: Combinations of Functions

    Oh of course, I don't know why I didn't think of that. So then:
    x-4 = 9x-8
    x=1/2

    And inputting that value of x into the equations they are equal. So that is right? Now I'm still fairly confused for the first question and don't know where to begin with it.
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