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Math Help - domain and range of inverse trig functions

  1. #1
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    Talking domain and range of inverse trig functions

    hey, can someone tell me how to do these? like explain it and all

    Find the domain and range of:

    a)  y = 2tan^{-1}x

    do for y = 2tanx.. the domain would be -pi/2 < x < pi/2
    the range is all real y

    so for  y = 2tan^{-1}x to domain should be all real x
    and range should be -pi/2 < x < pi/2

    But the book said that the range should be -pi< x < pi
    I don't get why.

    b)  y=2cos^{-1}x - 1

    I don't understand this one at all


    THANKS IN ADVANCE
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  2. #2
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    Quote Originally Posted by differentiate View Post
    hey, can someone tell me how to do these? like explain it and all

    Find the domain and range of:

    a)  y = 2tan^{-1}x

    do for y = 2tanx.. the domain would be -pi/2 < x < pi/2
    the range is all real y

    so for  y = 2tan^{-1}x to domain should be all real x
    and range should be -pi/2 < x < pi/2 Mr F says: There's a dilation by a factor of 2 in the vertical direction therefore the range is {\color{red}2 \left(-\frac{\pi}{2}\right) < x < 2 \left(\frac{\pi}{2}\right) \Rightarrow -\pi < x < \pi}.

    But the book said that the range should be -pi< x < pi
    I don't get why.

    b)  y=2cos^{-1}x - 1

    I don't understand this one at all


    THANKS IN ADVANCE
    For (b) is it  y=2cos^{-1}(x) - 1 or  y=2cos^{-1}(x - 1) ?
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  3. #3
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    you need to find domain of:

    g(x) = 2tan^{-1}(x)

    if we have any function ( f) that have inverse function ( f^{-1}) that means that f is bijection ( 1 - 1 )

    or:

    f: A -> B -----> f^{-1}: B -> A and card( A ) = card( B )

    ( card ---> cardinal number or number of elements in set )

    so if I know that:

    g(x) = 2tan^{-1}(x) = 2arctan(x)
    than it is easy to find g^{-1}(x)

    if I know this:

    f( x ) = tan(x)----> f^{-1}(x) = arctan(x)

    now i know that:

    g(x) = 2 arctan(x)----> g^{-1}(x) = \frac{tan(x)}{2}

    and i know that: R_f = \mathbb{R} ( R_f is range of tan )

    also I know:

    D_f = <\frac{-\pi}{2},\frac{\pi}{2}> ( D_f is domain of tan )

    so from:

    f: A -> B -----> f^{-1}: B -> A

    now i know that

    domain of 2arctan(x) is R,

    and i know that range of arctan(x) is <\frac{-\pi}{2},\frac{\pi}{2}>

    so range of 2arctan(x) is
    <-\pi,\pi>

    try now b)
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