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Math Help - still ont to ont problem

  1. #1
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    Smile still ont to ont problem

    (a). Define the function f:

    by f(x,y) = (x+2y, 3x+4y). Is f one-to-one?
    (Justify your answer)

    can i answer this question like follow
    x+2y=x+2y
    x+y=x+y

    3x+4y=3x+4y
    x+y=x+y

    ?i know sounds stupid but this is all i can remember from the lecture.
    anyone can give me some ideas? thanks^0^

    one to one problem~~~... i mean
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  2. #2
    Super Member Matt Westwood's Avatar
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    Have you studied matrices?

    What you can do is express the simultaneous equations in matrix form. As it's a simple 2x2 it's then straightforward to determine whether the determinant is zero.

    If it is not zero, then the matrix is invertible, and therefore the function is one-to-one and onto.

    Any help?
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  3. #3
    MHF Contributor

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    Quote Originally Posted by paulzho89 View Post
    (a). Define the function f:

    by f(x,y) = (x+2y, 3x+4y). Is f one-to-one?
    (Justify your answer)

    can i answer this question like follow
    x+2y=x+2y
    x+y=x+y

    3x+4y=3x+4y
    x+y=x+y

    ?i know sounds stupid but this is all i can remember from the lecture.
    anyone can give me some ideas? thanks^0^
    Do you remember the definition of "one to one function"? A function is "one to one" if and only if f(a)= f(b) only when a= b. Since here, f is applied to two numbers, that is "f(a,b)= f(x,y) if and only if (a,b)= (x,y)" which is the same as saying a= x and b= y.

    From f(a,b)= (a+ 2b, 3a+ 4b)= (x+ 2y, 3x+ 4y)= f(x,y), which is the same as saying a+ 2b= x+ 2y and 3a+ 4b= 3x+ 4y, can you show that a= x and b= y?
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    Do you remember the definition of "one to one function"? A function is "one to one" if and only if f(a)= f(b) only when a= b. Since here, f is applied to two numbers, that is "f(a,b)= f(x,y) if and only if (a,b)= (x,y)" which is the same as saying a= x and b= y.

    From f(a,b)= (a+ 2b, 3a+ 4b)= (x+ 2y, 3x+ 4y)= f(x,y), which is the same as saying a+ 2b= x+ 2y and 3a+ 4b= 3x+ 4y, can you show that a= x and b= y?
    thanks ..help me out^.^
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