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Math Help - Graph each function and its inverse

  1. #1
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    Smile Graph each function and its inverse

    Learning how to do these are the last things I need to learn for today.

    Graph each function and its inverse.

    1) F(x) = |x - 3| + 2


    2) F(x) = (x + 2)2 + 3
    (The small 2 represents power)
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  2. #2
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    The first function is an inverse of the second and vice versa. Simply replace the x's in your function with y's to receive the inverse function.

    f(x) = (x + 2)^2 + 3
     y = (x + 2)^2 + 3
     x = (y + 2)^2 + 3
     -(y+2)^2 = -x + 3
    (y + 2)^2 = x - 3
    y + 2 = \sqrt{x - 3}
    y = \sqrt{x - 3} + 2


    Final Inverse of f(x) = (x + 2)^2 + 3 :

    f(x) = \sqrt{x - 3} + 2
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  3. #3
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    Question

    Quote Originally Posted by Failbait View Post
    y + 2 = \sqrt{x - 3}
    y = \sqrt{x - 3} + 2
    Wouldn't the 2 change to a -2 when it switches from the side with the Y to the side with the square root of X - 3?
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  4. #4
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    Ah yes, sorry about that.
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  5. #5
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    How would I solve the first problem?

    Graph each function and its inverse.

    1) F(x) = |x - 3| + 2
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Brazuca View Post
    How would I solve the first problem?

    Graph each function and its inverse.

    1) F(x) = |x - 3| + 2
    recall that for absolute values you have to consider two cases, when what's inside absolute value signs is negative and when it is positive.

    to find the inverse, we switch x and y and solve for y:

    f(x) = |x - 3| + 2 = \left \{ \begin{array}{cc} x - 1, & \mbox { if } x \ge 3 \\ 5 - x, & \mbox { if } x < 3 \end{array} \right.

    work on each piece separately.


    For the first graph:

    y = x - 1 for x \ge 3

    For inverse, switch x and y:

    x = y - 1 for y \ge 3

    \Rightarrow y = x + 1 for y \ge 3 (which is equivalent to x \ge 2)


    now for the second graph:

    y = 5 - x for x < 3

    For inverse, switch x and y:

    x = 5 - y for y < 3

    \Rightarrow y = 5 - x for y < 3 (which is equivalent to x > 2)


    thus, putting these together we have:

    f^{-1}(x) = \left \{ \begin{array}{cc} x + 1, & \mbox { if } x \ge 2 \\ 5 - x, & \mbox { if } x > 2 \end{array} \right.


    the graph is below. the green is f(x), the red is f^{-1}(x)
    Attached Thumbnails Attached Thumbnails Graph each function and its inverse-abs.jpg  
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