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Math Help - Find a Function G and F[x]

  1. #1
    Junior Member Godfather's Avatar
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    Find a Function G and F[x]

    Let f[x]=(x^2)+1. Find a function g so that
    [a] [fg][x]=(x^4)-1
    [b] [f+g][x]=3(x^2)
    [c] [f/g][x]=1
    [d] f[g[x]]=9(x^4)+1
    [e] g[f[x]]=9(x^4)+1
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Godfather View Post
    Let f[x]=(x^2)+1. Find a function g so that
    [a] [fg][x]=(x^4)-1
    Hint: note that if we factored this, it would be the difference of two squares, with f being one factor

    [b] [f+g][x]=3(x^2)
    we want x^2 + 1 + g(x) = 3x^2

    simply solve for g(x)


    [c] [f/g][x]=1
    Hint: any number divided by itself is 1

    [d] f[g[x]]=9(x^4)+1
    we want (g(x))^2 + 1 = 9x^4 + 1

    solve for g(x)


    [e] g[f[x]]=9(x^4)+1
    i can't really think of a nice hint to give you here...

    g(x) = (3x - 3)^2 + 1
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  3. #3
    Junior Member Godfather's Avatar
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    Hint: any number divided by itself is 1

    we want (g(x))^2 + 1 = 9x^4 + 1

    solve for g(x)
    So it would be X^2 +1 but what is x?

    i can't really think of a nice hint to give you here...

    g(x) = (3x - 3)^2 + 1[/quote]
    How did u get that?

    What are all of the x's after [f/g][x]
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    your post makes no sense. separate my quotes from your questions to make sure i don't miss anything.

    Quote Originally Posted by Godfather View Post
    Hint: any number divided by itself is 1
    you said nothing after you quoted this, so i guess you got the answer

    (to the f(g(x)) question: So it would be X^2 +1?
    no. and you can check this. if g(x) = x^2 + 1 and f(x) = x^2 + 1, then f(g(x)) = \left( x^2 + 1 \right)^2 + 1 = x^4 + 2x^2 + 2 which is not 9x^4 + 1

    but what is x?
    do you not know function notation? the x's here just denote that x is the variable. so saying f(x) means that we have a function of x, that is, a function in which the variable is x.

    How did u get that?
    like i said, i don't know a nice hint to tell you here. but i essentially tried to get f(x) out of g(x). that is, i rewrote g(x) with the purpose of somehow giving rise to the formula for f(x)

    we started with g(x) = 9x^4 + 1

    now, 9x^4 + 1 = \left( 3x^2 \right)^2 + 1

    = \left[ 3 \left( x^2 {\color{blue} + 1 - 1} \right) \right]^2 + 1 ...........note that i added and subtracted one from the x^2 factor, this is like adding zero, so i'm not changing anything.

    = \left[ 3 { \color{red} \left( x^2 + 1 \right) } - 3 \right]^2 + 1 ...now do you see why i added and subtracted 1? when i multiplied out the -1, i am left with f(x) is the brackets. this is what i was after

    = \left[ 3 { \color{red} \left( f(x) \right) } - 3 \right]^2 + 1

    so now, i just let g(x) be the function that has f(x) plugged into it

    What are all of the x's after [f/g][x]
    i answered this already
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