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    It states in my book that

    "A surjective function f has an inverse if and only if its graph is cut at most once by any horizontal line."

    What i don't quite understand here is how a surjective function has an inverse if it's cut at most once by a horizontal line. Surely then by definition then it's not a surj..............


    Realising that f(x)=x³ is an example are there any other examples of such a function which fits this definition?

    And is an injective function invertible provided they are increasing or decreasing like x^5 for example? or -x^3
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    If  f is surjective and any horizntal line cuts the graph at at most one point then this means that  f is injective and so this means it is bijective. A function has an inverse if and only if it is bijective.
    A function that is strictly increasing or strictly decreasing is injective (assuming function is continuous) but this does not mean necessarily that the function is surjective and so it may not be bijective so may not have an inverse.
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    (Original post by will'o'wisp2)
    It states in my book that
    What book are you using? I've not come across these horizontal/vertical line tests spelt out explicitly .
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    (Original post by B_9710)
    If  f is surjective and any horizntal line cuts the graph at at most one point then this means that  f is injective and so this means it is bijective. A function has an inverse if and only if it is bijective.
    A function that is strictly increasing or strictly decreasing is injective (assuming function is continuous) but this does not mean necessarily that the function is surjective and so it may not be bijective so may not have an inverse.
    You don't need to assume continuity for strictly increasing => iniective.
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    RULER TEST gwm
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    (Original post by B_9710)
    If  f is surjective and any horizntal line cuts the graph at at most one point then this means that  f is injective and so this means it is bijective. A function has an inverse if and only if it is bijective.
    A function that is strictly increasing or strictly decreasing is injective (assuming function is continuous) but this does not mean necessarily that the function is surjective and so it may not be bijective so may not have an inverse.
    I thought it had one if it was injective of course bijective is included
    I mean i think that if it's bijective then it's invertible
    (Original post by ghostwalker)
    What book are you using? I've not come across these horizontal/vertical line tests spelt out explicitly .
    The custom book my uni gives out made by some of the lecturers who teach me.

    I've also just realised i have no idea what it means to have an "invertible function" cus google gives me inverse and did you mean inverse function -__-
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    (Original post by will'o'wisp2)
    I thought it had one if it was injective of course bijective is included
    I mean i think that if it's bijective then it's invertible
    Take f: \mathbb{R}_{\geqslant 0} \to \mathbb{R} given by x\mapsto x^2 then this is injective but not surjective and has no inverse. If it's invertible then it is bijective and if it is bijective then it is invertible.


    I've also just realised i have no idea what it means to have an "invertible function" cus google gives me inverse and did you mean inverse function -__-
    The inverse to a function f:X \to Y is a function g: \mathrm{im} \,f \to X with gf=fg = \mathrm{id}. A function f is called invertible if an inverse function exists. [Of course, here \mathrm{im} \, f = Y ]
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    (Original post by Zacken)
    Take f: \mathbb{R} \to \mathbb{R} given by x\mapsto x^2 then this is injective but not surjective and has no inverse. If it's invertible then it is bijective and if it is bijective then it is invertible.




    The inverse to a function f:X \to Y is a function g: \mathrm{im} \,f \to X with gf=fg = \mathrm{id}. A function f is called invertible if an inverse function exists. [Of course, here \mathrm{im} \, f = Y ]
    I thought injective meant 1 to 1 but for example -1 and 1 both real numbers give out the same output so surely that's just neither?

    Thanks man
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    (Original post by will'o'wisp2)
    I thought injective meant 1 to 1 but for example -1 and 1 both real numbers give out the same output so surely that's just neither?

    Thanks man
    1-to-1 doesn't really mean anything, but I believe the majority of people use 1-to-1 to mean bijective.

    Yeah sorry, I meant to do f:\mathbb{R}_{\geqslant 0} \to \mathbb{R} given by x \mapsto x^2
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    (Original post by Zacken)
    1-to-1 doesn't really mean anything, but I believe the majority of people use 1-to-1 to mean bijective.

    Yeah sorry, I meant to do f:\mathbb{R}_{\geqslant 0} \to \mathbb{R} given by x \mapsto x^2
    Oh do they? Ah that's why. For the course i do bijective is defined as a function which is both injective and surjective .-.
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    (Original post by will'o'wisp2)
    Oh do they? Ah that's why. For the course i do bijective is defined as a function which is both injective and surjective .-.
    Yes, that's the definition of bijective.
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    (Original post by Zacken)
    1-to-1 doesn't really mean anything, but I believe the majority of people use 1-to-1 to mean tbijective.
    Possibly this has changed, but I have always understood 1-to-1 to be synonymous with injective (and onto synonymous with surjective).

    Wikipedia seems to agree with me, FWIW.
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    (Original post by DFranklin)
    Possibly this has changed, but I have always understood 1-to-1 to be synonymous with injective (and onto synonymous with surjective).

    Wikipedia seems to agree with me, FWIW.
    In hindsight, you're right. I think what has me confused is that people use 1-to-1 for injective and 1-to-1 correspondence for bijective. I tend to try and avoid all those terms and just stick to injective/bijective myself (although I do use onto).
 
 
 
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