Masih tentang Involusi

Buktikan bahwa fungsi berikut merupakan involusi:

  1. g(x) = (kx3)1/3.
  2. f(x) = –ln[tanh(x/2)].

Catatan. Di bawah ini adalah grafik fungsi f(x) = –ln[tanh(x/2)].

fungsi involusi

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Bandung, 19-05-2017

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3 comments

  1. 1. Misal y = g(x) = (k – x^3)^1/3
    Maka y^3 = k – x^3
    x^3 = k – y^3
    x = (k – y^3)^1/3
    shg g-1(x) = (k – x^3)^1/3 = g(x)
    Jadi g(x) adl fungsi involusi.

    2. tanh x/2 = (e^x – 1)/(e^x + 1)
    Misal y = f(x) = -ln((e^x – 1)/(e^x + 1))
    Maka – y = ln((e^x – 1)/(e^x + 1))
    (e^x – 1)/(e^x + 1) = e^-y
    e^x = (e^-y + 1)/(- e^-y + 1)
    e^-x = (- e^-y + 1)/(e^-y + 1)
    e^-x = (e^-y – 1)/(e^y + 1)
    – x = ln((e^-y – 1)/(e^y + 1))
    shg f-1(x) = -ln((e^-y – 1)/(e^y + 1)) = -ln(tanh x/2) = f(x)
    Jadi f(x) adl fungsi involusi.

    Like

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