What is the inverse of f(x) = x + sqrt(x)?

No matter what I end up with x^2 = y^2 + y and have no idea how to get it to just one y.

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  • Harish
    Lv 7
    6 years ago
    Favourite answer

    f(x) = x + √x

    1.

    To find the inverse, first change to x, y notation.

    y = x + √x

    Then interchange the positions of x and y and solve for y.

    x = y + √y

    The solution for the new y is the inverse function.

    x^2 = y^2 + y + 2y^(3/2)

    x =

  • Anonymous
    6 years ago

    ƒ(x) = x + √x

    y - x = √x

    y² - 2xy + x² = x

    x² + (-2y - 1)x + y² = 0

    Invoke quadratic equation:

    x = 2y + 1 ± √[4y² + 4y + 1 - 4y²] ÷ 2

    x = 2y + 1 ± √[4y + 1] ÷ 2

    Using the original equation, if x = 1, y = 2. Subbing in y = 2 into the inverse gets 4 if we use the positive root or 1 if we use the negative. Therefore, the negative root is the right one.

    ƒ⁻¹(x) = x - 1/2 √[4x + 1] + 1/2

  • ted s
    Lv 7
    6 years ago

    DON'T know how you got x² = y² + y...you appear to use [ a + b]² = a² + b² !!!

    solve y = x + √x for x....[ y - x ]² = x = y² - 2xy + x² ---> x² + x [ - 2y -1 ] + y² = 0...

    quadratic equation yields x = { (2y + 1) -√ [ (2y + 1)² - 4 y² ] } / 2...

    if you demand the dependent variable be y then interchange x & y..

    note : no ± since when y = 0 so must x = 0

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