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Thread: 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 $\displaystyle x$'s in your function with $\displaystyle y$'s to receive the inverse function.

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


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

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

    Quote Originally Posted by Failbait View Post
    $\displaystyle y + 2 = \sqrt{x - 3}$
    $\displaystyle y = \sqrt{x - 3} + 2$
    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:

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

    work on each piece separately.


    For the first graph:

    $\displaystyle y = x - 1$ for $\displaystyle x \ge 3$

    For inverse, switch x and y:

    $\displaystyle x = y - 1$ for $\displaystyle y \ge 3$

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


    now for the second graph:

    $\displaystyle y = 5 - x$ for $\displaystyle x < 3$

    For inverse, switch x and y:

    $\displaystyle x = 5 - y$ for $\displaystyle y < 3$

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


    thus, putting these together we have:

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


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