If f(x) =4×2 – 8x – 20 and g(x) = 2x + a, find the value of a so that the y-intercept of the graph of the composite function (fog)(x) is (0,

Question

If f(x) =4×2 – 8x – 20 and g(x) = 2x + a, find the value of a so that the y-intercept of the graph of the composite function (fog)(x) is (0, 25).

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Maris 6 months 2021-08-02T15:55:20+00:00 1 Answers 4 views 0

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    2021-08-02T15:56:39+00:00

    Answer:

    The possible values are a = -2.5 or a = 4.5.

    Step-by-step explanation:

    Composite function:

    The composite function of f(x) and g(x) is given by:

    (f \circ g)(x) = f(g(x))

    In this case:

    f(x) = 4x^2 - 8x - 20

    g(x) = 2x + a

    So

    (f \circ g)(x) = f(g(x)) = f(2x + a) = 4(2x + a)^2 - 8(2x + a) - 20 = 4(4x^2 + 4ax + a^2) - 16x - 8a - 20 = 16x^2 + 16ax + 4a^2 - 16x - 8a - 20 = 16x^2 +(16a-16)x + 4a^2 - 8a - 20

    Value of a so that the y-intercept of the graph of the composite function (fog)(x) is (0, 25).

    This means that when x = 0, f(g(x)) = 25. So

    4a^2 - 8a - 20 = 25

    4a^2 - 8a - 45 = 0

    Solving a quadratic equation, by Bhaskara:

    \Delta = (-8)^2 - 4(4)(-45) = 784

    x_{1} = \frac{-(-8) + \sqrt{784}}{2*(4)} = \frac{36}{8} = 4.5

    x_{2} = \frac{-(-8) - \sqrt{784}}{2*(4)} = -\frac{20}{8} = -2.5

    The possible values are a = -2.5 or a = 4.5.

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