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Math Help - 3 intercepts between f(x) and inverse??

  1. #1
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    Unhappy 3 intercepts between f(x) and inverse??

    Something I find really odd.... A question provides us with a square root function, and asks us to sketch it and its inverse. Strangely enough, f(x) and its inverse intersect not only along y=x but at two other points.

    My question is, simply, how do you figure out how many intersections there will be between a function and its inverse? Even my teacher was perplexed.
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    Quote Originally Posted by freswood
    Something I find really odd.... A question provides us with a square root function, and asks us to sketch it and its inverse. Strangely enough, f(x) and its inverse intersect not only along y=x but at two other points.

    My question is, simply, how do you figure out how many intersections there will be between a function and its inverse? Even my teacher was perplexed.
    You have, f(x)=\sqrt{x} since this is a bijective function it have an inverse, y=x^2 \mbox{ for } x\geq 0. To find intersection points you need to consider,
    \sqrt{x}=x^2 for x\geq 0
    Square both sides,
    x=x^4
    Thus,
    x^4-x=0
    Thus,
    x(x^3-1)=0
    Thus,
    x=0 and x^3-1=0
    All the real solutions are,
    x=0,1

    I do not see your problem?
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  3. #3
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    In the example they gave, it was:

    f(x) = 4 - 2[root](2x+6)

    We've always been taught that a function and its inverse will intersect on the line y=x, but never any mention of anything else.
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    Quote Originally Posted by freswood
    We've always been taught that a function and its inverse will intersect on the line y=x, but never any mention of anything else.
    What is that supposed to mean?!?!
    A function and its inverse are line relfections in line y=x that is probably what you mean. Not that they intersect at all point on y=x
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    Quote Originally Posted by ThePerfectHacker
    What is that supposed to mean?!?!
    A function and its inverse are line relfections in line y=x that is probably what you mean. Not that they intersect at all point on y=x
    In order for a graph to intersect its inverse we have to have the following situation: Let y = f(x) be the original graph. The inverse function is represented by x = f(y), or y = g(x). The only way these will intersect is if there is a point (x,y) in the original graph and a point (y,x) in the inverse graph such that (x,y)=(y,x), ie. x = y. Thus all such intersection points will be on the line y = x.

    (BTW: The graph of a square root function is essentially the graph of part of a parabola either opening to the right or left. So I am going to refer to your square root function as a parabola. In the following paragraph, then, the "parabola" mentioned can either be your square root function, or its inverse.)

    Now, we are looking for the set of intersection points of a parabola, its inverse, and the line y = x. A parabola can intersect with a line in AT MOST 2 points. These are the same two points that the inverse function will cross the line y = x at. So your solution set of the crossing points of your parabola and its inverse is at most 2 points, not 3.

    I must therefore conclude that something was either solved or graphed incorrectly for you to have 3 points.

    -Dan
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    Quote Originally Posted by freswood
    Something I find really odd.... A question provides us with a square root function, and asks us to sketch it and its inverse. Strangely enough, f(x) and its inverse intersect not only along y=x but at two other points.

    My question is, simply, how do you figure out how many intersections there will be between a function and its inverse? Even my teacher was perplexed.
    Hello,

    in addition to all posts I only want to point out, that you have to look thoroughly at the domain and range of a function and its inverse.

    In your case:

    f(x)=4-\sqrt{2x+6}\ \wedge \ f(x) \leq 4 you'll get the inverse:

    g(x)=\frac{1}{8} x^2-x-1\ \wedge \ x\leq4

    If you take g without the restriction you'll get indeed 3 intercepting points.

    Greetings

    EB
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  7. #7
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    Quote Originally Posted by freswood
    Something I find really odd.... A question provides us with a square root function, and asks us to sketch it and its inverse. Strangely enough, f(x) and its inverse intersect not only along y=x but at two other points....
    Hello,

    it's me again.

    I've attached a diagram, which shows that your teacher is right.

    Greetings

    EB
    Attached Thumbnails Attached Thumbnails 3 intercepts between f(x) and inverse??-fkt_inv_3interc.gif  
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    Quote Originally Posted by earboth
    Hello,

    it's me again.

    I've attached a diagram, which shows that your teacher is right.

    Greetings

    EB
    Thanks for clarifying it. So how would I know (without using a calculator) whether a graph intersects its inverse more than once? Our end of year exam has a part with no calc.
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  9. #9
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    Quote Originally Posted by freswood
    Thanks for clarifying it. So how would I know (without using a calculator) whether a graph intersects its inverse more than once? Our end of year exam has a part with no calc.
    I don't know if this is of any help, but the number of points of intersection
    of a function f and its inverse f^{-1} is equal to the number of roots of
    f(f(x))=x where x is in the domain of both f and f^{-1}.

    This is of course equivalent to saying it is equal to the number of roots of
    f(x)=f^{-1}(x) in the intersection of the two domains, so its not saying
    anything new really.

    RonL
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  10. #10
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by earboth
    Hello,

    it's me again.

    I've attached a diagram, which shows that your teacher is right.

    Greetings

    EB
    My apologies. I was obviously only considering symmetric points!

    -Dan
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