Anonymous
Anonymous asked in Science & MathematicsMathematics · 8 months ago

Someone help me with this math question?

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3 Answers

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  • 8 months ago

    g^-1(x)=(5-kx)/3

    =>

    g(x)=(5-3x)/k

    g(x^2)=2f(-x)

    =>

    [5-3(x^2)]/k=2(3x^2-5)

    =>

    5-3x^2=6kx^2-10k---(1)

    (k=/=0)

    =>

    x^2=(10k+5)/(6k+3)

    =>

    x=+/-sqr[(10k+5)/(6k+3)]

    =>

    (10k+5)/(6k+3)>=0

    =>

    k>-1/2 or k<-1/2

    [Note that k=-1/2=>

    (10k+5)/(6k+3)=0/0 which

    is uncertain; but (1) holds

    =>k=-1/2 is acceptable]

    =>

    k can be any real number

    but 0, or write k=R-{0}.

    • Pinkgreen
      Lv 7
      8 months agoReport

      The question is ambiguous, it may be an equation of x & k
      is a parameter. So it must be stated clearly that it is an identity in x.

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  • Ian H
    Lv 7
    8 months ago

    f(x) = 3x^2 – 5

    2f(-x) = 2[3(-x)^2 – 5] = 6x^2 – 10 ........(1)

    The next part assumes you meant inverse function rather than reciprocal.

    The inverse of an inverse is the original function, so to find inverse of

    Let g^-1(x) = y(x) = (5 – kx)/3

    x = (5 – 3y)/k ....now swap roles

    y = (5 – 3x)/k and interpret role-reversed as

    g(x) = (5 – 3x)/k

    g(x^2) = (5 – 3x^2)/k .........................(2)

    For g(x^2) = 2f(-x) we have

    (5 – 3x^2)/k = 6x^2 – 10

    k = -(3x^2 - 5)/(6x^2 – 10) = -1/2

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  • Alan
    Lv 7
    8 months ago

    g^-1(x) = (5 -kx)/3

    f(x) = 3x^2 - 5

    g(x^2) = 2*f(-x)

    so 2*f(-x) = 2* (3x^2 -5)

    since (-x)^2 =x^2

    2*f(-x) = 6x^2 -10

    g^-1(x) = (5-kx)/3

    so inverse of the inverse is

    the original function

    so if inverse is:

    y = (5-kx)/3

    swap x and y and solve for new y

    x= (5-ky)/3

    3x = 5-ky

    ky= 5-3x

    y = (5-3x)/k

    g(x) = (5-3x)/k

    g(x^2) = (5-3x^2)/k

    so you want k such that

    6x^2 -10 = (5-3x^2)/ k

    6kx^2 -10k = 5- 3x^2

    Equate like terms

    6kx^2 = -3x^2

    divide both sides by 3x^2

    2k = -1

    k = -1/2

    checking

    -10k = 5

    divide both sides by -10

    k = -5/10 = -1/2 (good ,

    both answers must be the same)

    so k = -1/2

    checking the answer

    g^(-1)(x) = (5+(x/2))/ 3

    g^-1(x) = 5/3 + x/6

    inverse of this

    x= 5/3 +y/6

    multiply by 6

    6x = 10 + y

    y = 6x-10

    g(x) = 6x -10

    g(x^2) = 6x^2 -10

    f(x) =3x^2-5

    2*f(-x) = 2*(3(-x)^2 -5) = 6x^2 -10

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