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Math Help - Quick question about inverse functions.

  1. #1
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    Quick question about inverse functions.

    Under what conditions is the inverse of a function a function?

    Can/does this have something to do with the input and outputs of a function or is it as simple whether or not it passes the vertical line test?

    Thanks!


    -Tom
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  2. #2
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    Hello, Tom!

    Under what conditions is the inverse of a function a function?

    Can/does this have something to do with the input and outputs of a function
    or is it as simple whether or not it passes the vertical line test?

    It is as simple as the vertical line test.


    There's an "eyeball" approach, too.

    Look at the graph of the function f(x).
    If it passes the "horizontal line test", its inverse is a function.
    (That is, if all horizontal lines intersect the graph in at most one point,
    . . then the inverse is also a function.)

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    Thank you!
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    Quote Originally Posted by Soroban View Post
    Hello, Tom!


    It is as simple as the vertical line test.


    There's an "eyeball" approach, too.

    Look at the graph of the function f(x).
    If it passes the "horizontal line test", its inverse is a function.
    (That is, if all horizontal lines intersect the graph in at most one point,
    . . then the inverse is also a function.)

    Here's another quick question... So an example of that would be the functions (1,2) , (3,4). That's a function. The inverse is (2,1) , (4,3) and that's a function too. Am I getting this correctly?

    Thanks,


    -Tom
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    Quote Originally Posted by TomCat View Post
    Here's another quick question... So an example of that would be the functions (1,2) , (3,4). That's a function. The inverse is (2,1) , (4,3) and that's a function too. Am I getting this correctly?

    Thanks,


    -Tom
    those are coordinates.

    a function is an equation, such as "f(x)=(1/2)x+5"
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    Quote Originally Posted by TomCat View Post
    Here's another quick question... So an example of that would be the functions (1,2) , (3,4). That's a function. The inverse is (2,1) , (4,3) and that's a function too. Am I getting this correctly?

    Thanks,


    -Tom
    Yes. A function (more formally but not completely) is a set of ordered pairs. Sometimes you can infinite many like (x,x^2) or sometime finite like here. Thus, yes it is a function and that is its inverse.
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  7. #7
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    Quote Originally Posted by ThePerfectHacker View Post
    Yes. A function (more formally but not completely) is a set of ordered pairs. Sometimes you can infinite many like (x,x^2) or sometime finite like here. Thus, yes it is a function and that is its inverse.
    really? Do you mean that any two points can have a function, or that any two points are a function.
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  8. #8
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    Quote Originally Posted by Quick View Post
    really? Do you mean that any two points can have a function, or that any two points are a function.
    A single point can define a function.
    For example, {(1,1)}

    There is a formal definition of a function that appeared in the 19th Century due to one of my favorites, Dirichlet. Ever since then people started using his definition. A single point fits this definition.

    The first time the word "function" was used by the German polymath/mathematician Gottfiend von Leibniz (he was as much as a nemesis to Newton as I am to CaptainBlank). He definied informally as a relationship between two things. For example, the size of a tree as a function of time.

    When LaTeX is back online I can give you a formal treatement of what a function is in a mathematical sense. (But be warned this is going to be your first tour of pure mathematics. It might be dangerous).
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  9. #9
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    Quote Originally Posted by ThePerfectHacker View Post
    When LaTeX is back online I can give you a formal treatement of what a function is in a mathematical sense.
    See you in two weeks
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