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Math Help - 1-1 function

  1. #1
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    1-1 function

    Hi,

    I have this question:
    if f and f o g are 1 to 1 functions, does it follow that g is 1-1 function?

    I tried for example ,

    if g(3)=4 and g(7)=4,
    then f( g(3) )= f( G(7) )=f(4) will produce the 1-1 function property.( it does not work)

    I have tried other things but still , it doesn't work can someone help me?

    B
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  2. #2
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    Let,
    f:Y--> Z be one-to-one
    g:X--> Y be a function
    f o g: X-->Z be one-to-one
    Let,
    h = f o g
    Then,
    we need to show that if g(x1)=g(x2) then x1=x2
    We know that if,
    h(x1)=h(x2) then x1=x2 thus,
    f(g(x1))=f(g(x2))
    then,
    g(x1)=g(x2)
    I cannot see how you can conclude that g is one-to-one from these statements.
    I did not think of a counterexample but I would guess no.
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  3. #3
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    Actually g has to be one-to-one.
    Suppose that g(a)=g(b).
    The by function definition f(g(a))=f(g(b)) or fog(a)=fog(b).
    But we are given that fog is one-to-one, therefore a=b.
    If g(a)=g(b) then a=b; g is one-to-one.
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  4. #4
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    Quote Originally Posted by Plato View Post
    Actually g has to be one-to-one.
    Suppose that g(a)=g(b).
    The by function definition f(g(a))=f(g(b)) or fog(a)=fog(b).
    But we are given that fog is one-to-one, therefore a=b.
    If g(a)=g(b) then a=b; g is one-to-one.
    Good explanation.
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