Results 1 to 13 of 13

Math Help - How to prove x^2 is a function?

  1. #1
    Newbie
    Joined
    Oct 2011
    Posts
    9

    How to prove x^2 is a function?

    Title says it all. How do you prove x^2 is a function *besides* the vertical line test?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5

    Re: How to prove x^2 is a function?

    Quote Originally Posted by vandrop View Post
    Title says it all. How do you prove x^2 is a function *besides* the vertical line test?
    You need to know what the definition of a function is. Then, you need to consider what short-cut the vertical line test provides in applying the definition.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member TheChaz's Avatar
    Joined
    Nov 2010
    From
    Northwest Arkansas
    Posts
    600
    Thanks
    2

    Re: How to prove x^2 is a function?

    Quote Originally Posted by mr fantastic View Post
    You need to know what the definition of a function is. Then, you need to consider what short-cut the vertical line test provides in applying the definition.
    Indeed... x^2 is the example used in the wikipedia page on functions !
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,386
    Thanks
    752

    Re: How to prove x^2 is a function?

    the formal definiton of a function is: f(x) ≠ f(y) --> x ≠ y (well more specifically, (x1,x2) in f, and (x1,x3) in f -->x2 = x3, but these two statements are equivalent).

    so suppose that x^2 ≠ y^2. this means that x^2 - y^2 ≠ 0. hence (x + y)(x - y) ≠ 0.

    in particular NEITHER x + y nor x - y can be 0. so x - y ≠ 0, and thus x ≠ y.

    in the other form, suppose (x,x^2) and (x,y^2) are in f = {(a,b) in R x R : b = a^2}. then y^2 = x^2.

    it is sometimes helpful to see why some relations fail to be functions. suppose g = {(a,b) in R x R : a = 1}.

    (this is the vertical line x = 1). well, we have (1,2) in g, and (1,3) in g, but clearly 2 ≠ 3.

    therefore, we have no idea what value to assign to g(1), we have "too many choices".

    this is why we write f(a) for the value of a function f at the point a, there is only one possible number we could pick, if f is a function.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Oct 2008
    From
    Maine
    Posts
    70

    Re: How to prove x^2 is a function?

    A function essentially is saying that whenever you input a number for x you only get one answer for y. Anything else would be illogical.

    I view it like a coffee machine. When I put the coffee grinds and water in, I expect to get coffee every time. Not coffee the first time, orange juice the second and hot chocolate the third time. If that starts happening my machine is not functioning properly.

    Same thing with function equations:

    y = x^2

    If I am inputting 2 into this function then I should get a y=4 EVERY time. Otherwise something is wrong. It is logical to see this equation and realize that you cannot plug in a number for x and get two different answers. Not a formal proof, but a little easier to understand why it works worded this way I think.


    The non example Deveno was talking about referred to a vertical (up and down) line.

    If you take a look at the points (ordered pair) of a vertical line all of the x-coordinates (inputs) are the same and the outputs are different. You might see ordered pair like this:

    (1,2)
    (1,3)
    (1,4)
    etc

    So a vertical line is not a function because every time you put in 1 for x you get a different answer for y. This is illogical.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Oct 2008
    From
    Maine
    Posts
    70

    Re: How to prove x^2 is a function?

    A simple way to "prove" that y = x^2 is a function is that it moves from left to right forever and ever and does not stop its forward motion to go straight up or turn around. These qualities mean prove that x will never get you two different answers for y because you are always moving left or right so your x-coordinates will never repeat.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,603
    Thanks
    1421

    Re: How to prove x^2 is a function?

    That makes no sense to me at all. Functions do not "move" at all. I suspect you are talking about the graph of f(x) but it would be a good idea to say that! In any case, you are really using the "vertical line" test and vandrop said "*besides* the vertical line test".

    Suppose x^2= y_1 and x^2= y_2, with y_1\ne y_2. What can you say about x?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member
    Joined
    Oct 2008
    From
    Maine
    Posts
    70

    Re: How to prove x^2 is a function?

    To me the vertical line test is physically taking a vertical line and passing it along the graph and looking for it to cross two points of the line at the same time. Which is a different mindset than just looking to see if the line, curve or wave is always moving left to right without stopping/turning around and not using the vertical line to accomplish this.

    They are both essentially accomplishing the same thing so I guess I can see how they are thought of as the same thing.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Aug 2011
    Posts
    243
    Thanks
    56

    Re: How to prove x^2 is a function?

    How to prove x^2 is a function?
    Also can you answer to questions such as :
    - How to prove 3x+7 is a function?
    - How to prove x+7 is a function?
    - How to prove 7 is a function?
    - How to prove 0 is a function?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5

    Re: How to prove x^2 is a function?

    Quote Originally Posted by JJacquelin View Post
    Also can you answer to questions such as :
    - How to prove 3x+7 is a function?
    - How to prove x+7 is a function?
    - How to prove 7 is a function?
    - How to prove 0 is a function?
    Obviously either the vertical line test or the method outlined by HallsofIvy in post #7! I'd strongly advise the former given the level you appear to be studying at.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,386
    Thanks
    752

    Re: How to prove x^2 is a function?

    Quote Originally Posted by HallsofIvy View Post
    That makes no sense to me at all. Functions do not "move" at all. I suspect you are talking about the graph of f(x) but it would be a good idea to say that! In any case, you are really using the "vertical line" test and vandrop said "*besides* the vertical line test".

    Suppose x^2= y_1 and x^2= y_2, with y_1\ne y_2. What can you say about x?
    i'd like to point out that there is no difference between a function and its graph. that is, formally, a function is a certain kind of relationship between two sets,

    so the set of all points {(x,f(x)} which define f is what we mean (and draw) by the graph of f.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Member
    Joined
    Aug 2011
    Posts
    243
    Thanks
    56

    Re: How to prove x^2 is a function?

    Quote Originally Posted by mr fantastic View Post
    Obviously either the vertical line test or the method outlined by HallsofIvy in post #7! I'd strongly advise the former given the level you appear to be studying at.
    Ha! ha! Funny !
    But you didn't answer to the question "How to prove 0 is a function?"
    Of course, I know how to answer to this question. But You ?
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5

    Re: How to prove x^2 is a function?

    Quote Originally Posted by JJacquelin View Post
    Ha! ha! Funny !
    But you didn't answer to the question "How to prove 0 is a function?"
    Of course, I know how to answer to this question. But You ?
    If you know the answer to the questions you post, you're wasting the time of people who might reply in good faith, thinking they are helping someone.

    If you want to challenge members with a question (regardless of how trivial the question is), post the question here: http://www.mathhelpforum.com/math-help/f25/

    after reading the rules for posting in that subforum.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove a function is Onto
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: April 7th 2010, 08:01 AM
  2. prove function is one-to-one
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: February 8th 2010, 12:45 PM
  3. Function prove
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 26th 2009, 02:03 PM
  4. Prove that f(x) is an odd function?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 21st 2009, 02:09 PM
  5. odd function prove
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 21st 2009, 04:24 PM

Search Tags


/mathhelpforum @mathhelpforum