Find the inverse of the function and state its domain?

f(x) = 3sin^-1(x/4) -2 for -4<=x<=4

If you guys could help me (show work too) I would really appreciate it.

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  • 7 years ago
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    To find the inverse, consider f(x) as y, and switch every instance of x with y

    y = 3sin^-1(x / 4) - 2

    Inverse:

    x = 3sin^-1(y / 4) - 2

    And now solve for y:

    x+2 = 3sin^-1(y/4)

    (x+2)/3 = sin^-1(y/4)

    sin((x+2) / 3) = y/4

    4sin((x+2)/3) = y

    The domain of the inverse is equal to the range of the original (and vice versa)

    The range of the original (assuming it's in radians) would be:

    3sin^-1(-4/4) - 2 = 3*-pi/2 - 2

    3sin^-1(4/4) - 2 = 3*pi/2 - 2

    [-6.7124, 2.7124] (approximately)

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