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@sorumatikbot
!Ekran görüntüsü 2024-08-07 210600|672x500 [Link Silindi]

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Given f(x) = \frac{x-1}{x-2} and g(x) = x^2 . Solve for f(g(x)) = 5

Answer:

To solve for f(g(x)) = 5 , we first need to find an expression for f(g(x)) :

  1. Calculate g(x) :
    g(x) = x^2

  2. Substitute g(x) into f(x) :

    f(g(x)) = f(x^2) = \frac{x^2 - 1}{x^2 - 2}

  3. Set up the equation f(g(x)) = 5 :

    \frac{x^2 - 1}{x^2 - 2} = 5

  4. Solve for x :
    Multiply both sides by x^2 - 2 :

    x^2 - 1 = 5(x^2 - 2)
    x^2 - 1 = 5x^2 - 10

    Bring all terms to one side of the equation to set it to zero:

    x^2 - 1 - 5x^2 + 10 = 0
    -4x^2 + 9 = 0
    4x^2 = 9
    x^2 = \frac{9}{4}
    x = \pm \frac{3}{2}

  5. Check the solutions:

    • x = \frac{3}{2}
    • x = -\frac{3}{2}

The solution set is: \left\{ -\frac{3}{2}, \frac{3}{2} \right\} , which corresponds to option (I).

Therefore, the correct answer is (I) \left\{ -\frac{3}{2}, \frac{3}{2} \right\} .