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Math Help - Inverse fxn help

  1. #1
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    Inverse fxn help

    Let f(x)=(x^4)+(x^3)+1
    Let g(x) be the inverse of f(x) and define F(x)=f(2g(x)). Find an equation for the tangent line to y=F(x) at x=3.


    The answer the book gives is y=(88x-89)/7 I just have no clue how to get there! Please help!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Inverse fxn help

    Hint: g(3)=1 and g'(3)=1/f'(1) . Now, apply the Chain's Rule.
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  3. #3
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    Re: Inverse fxn help

    hello there
    in order to compute the tangent line u need
    the value of the function f(2g(x)) at x=3
    and the value of the derivative of the function
    at the same point meaning F'(3)
    first u have to find f(2g(x))
    just substitute in f(x) 2g(x)
    u get
    16g(x)^4+8g(x)^3+1
    now what we need is actually
    16g(3)^4+8g(3)^3+1
    to know the value of this expression we need
    to know the value of g(3)
    since g(x) is the inverse of f(x)
    then f(g(3))=3
    in other words:
    g(3)^4+g(3)^3+1=3
    g(3)^4+g(3)^3-2=0
    by guessing we get that the solution is:
    g(3)=1
    (there is another solution but its hard to get
    it actually we need approximation)
    now that we have g(3)=1
    then F(3)=16*1^4+8*1^3+1=25
    NOW we have to know the value of
    F'(3)(the derivative)
    F'(3)=4*16*g(3)^3*g'(3)+3*8*g(3)^2*g'(3)
    to now the value of F'(3) we have to
    know the value of g'(3) since we already know the value
    of g(3)
    now since f(g(x)=x
    if we derive both sides using the chain rule
    we get
    f'(g(x)*g'(x)=1
    now divide both sides by f'(g(x))
    g'(x)=1/f'(g(x)
    g'(x)=1/f'(g(x)
    g'(x)=1/(4g(x)^3+3g(x)^2)
    now just subsitute x=3
    u get
    g'(3)=1/7
    can u proceed from here??
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  4. #4
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    Re: Inverse fxn help

    Quote Originally Posted by islam View Post
    hello there
    in order to compute the tangent line u need
    the value of the function f(2g(x)) at x=3
    and the value of the derivative of the function
    at the same point meaning F'(3)
    first u have to find f(2g(x))
    just substitute in f(x) 2g(x)
    u get
    16g(x)^4+8g(x)^3+1
    now what we need is actually
    16g(3)^4+8g(3)^3+1
    to know the value of this expression we need
    to know the value of g(3)
    since g(x) is the inverse of f(x)
    then f(g(3))=3
    in other words:
    g(3)^4+g(3)^3+1=3
    g(3)^4+g(3)^3-2=0
    by guessing we get that the solution is:
    g(3)=1
    (there is another solution but its hard to get
    it actually we need approximation)
    now that we have g(3)=1
    then F(3)=16*1^4+8*1^3+1=25
    NOW we have to know the value of
    F'(3)(the derivative)
    F'(3)=4*16*g(3)^3*g'(3)+3*8*g(3)^2*g'(3)
    to now the value of F'(3) we have to
    know the value of g'(3) since we already know the value
    of g(3)
    now since f(g(x)=x
    if we derive both sides using the chain rule
    we get
    f'(g(x)*g'(x)=1
    now divide both sides by f'(g(x))
    g'(x)=1/f'(g(x)
    g'(x)=1/f'(g(x)
    g'(x)=1/(4g(x)^3+3g(x)^2)
    now just subsitute x=3
    u get
    g'(3)=1/7
    can u proceed from here??
    note that 'math' tags do not work ... use 'tex' tags for Latex
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