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algebra

Compute the domain of the real-valued function f(x)=sqrt(1-sqrt(2-x)). Thank you

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  1. first of all,
    2-x > 0
    -x > -2
    x < 2

    furthermore,
    1-√(2-x) > 0
    1 > √(2-x)
    √(2-x) < 1
    so 0 < 2-x < 1
    -2 < -x < -1
    2 > x > 1
    or 1 < x < 2

    so if x < 2 AND 1 < x < 2
    then

    1 < x < 2

    test:
    take a value outside that domain:
    let x = 5
    f(5) = √( 1 - √-3)) , not defined for the reals
    let x = .6
    f(.6) = √(1 - √1.4)
    = √(-.1832...) which is undefined as well

    but x = 1.3
    f(1.3) = √(1 - √.7)
    = √.16333997...) = a real number

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