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Math Help - Transformation - Inverse Question

  1. #1
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    Transformation - Inverse Question

    For the following functions

    Equation --> y=x^2-3

    a).Find Inverse f^-1
    b).graph f(x) and its inverse

    c).restric the domain of f so that f^-1 is also a function

    d). with the domain of f restricted, sketch a graph of f and f^-1

    k so what i did is first i did the inverse:

    f(x)=x^2+3
    so what i did is let

    x=y^2+3
    x-3=y^2
    y=+-sqrt/x-3

    Then i graphed the normal equation and it turned our fine, but i need help drawing the mirror image(with table of values and steps on how to do so)

    also someone help with part c and d in detial please

    Thank You
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  2. #2
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    Quote Originally Posted by usm_67 View Post
    For the following functions

    Equation --> y=x^2-3

    a).Find Inverse f^-1
    b).graph f(x) and its inverse

    c).restric the domain of f so that f^-1 is also a function

    d). with the domain of f restricted, sketch a graph of f and f^-1

    k so what i did is first i did the inverse:

    f(x)=x^2+3
    so what i did is let

    x=y^2+3
    x-3=y^2
    y=+-sqrt/x-3

    Then i graphed the normal equation and it turned our fine, but i need help drawing the mirror image(with table of values and steps on how to do so)

    also someone help with part c and d in detial please

    Thank You
    Hello,

    1. you somehow changed the term of the function. I'll take the first version for further considerations.

    If D is the domain of f and R is the range of f you got:

    f: y = x^2-3,~~D_f=\mathbb{R},~R_f=\{y|y\geq -3\}

    To calculate the inverse of f you change x and y and you have to change domain and range too:

    f^{-1}: x = y^2-3,~~D_{f^{-1}}=\{x|x\geq -3\},~R_{f^{-1}}=\mathbb{R} . As you can see the inverse of f is not a function.

    to c) Roughly speaking a continuously increasing (or decreasing) function has an inverse function too. Thus you have to divide f into two branches(?):
    The decreasing branch: f: y = x^2-3,~~D_f=\{x|x < 0\},~R_f=\{y|y\geq -3\}
    and the increasing branch: f: y = x^2-3,~~D_f=\{x|x \geq 0\},~R_f=\{y|y\geq -3\}

    Solve the equation of the inverse of for y and you get the functions and their inverses as:

    f_1: y = x^2-3,~~D_f=\{x|x < 0\},~R_f=\{y|y\geq -3\}~~ \Longrightarrow f_1^{-1}: y=-\sqrt{x+3},~ D_{f_1^{-1}}=\{x|x >-3\},~R_{f_1^{-1}}=\{y|y <0\}

    and

    f_2: y = x^2-3,~~D_{f_2}=\{x|x \geq 0\},~R_{f_2}=\{y|y\geq -3\}~~ \Longrightarrow f_2^{-1}: y=\sqrt{x+3},~ D_{f_2^{-1}}=\{x|x >-3\},~R_{f_1^{-1}}=\{y|y \geq 0\}

    I've attached a drawing. Corresponding parts of the funcrtion and it's inverse are marked in similar colours.
    Attached Thumbnails Attached Thumbnails Transformation - Inverse Question-funkt_umkehrfkt.gif  
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