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    How do you show functions are one to one

    eg

     f:x  -> (x+3)^{2}+1
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    I'd imagine you'd do it by performing a formula that my non-mathmatical mind can't conquer. ):
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    f isn't one-to-one. f(1)=f(-7)=17.
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    That function is not injective. (In other words, there are two distinct numbers x and y such that f(x) = f(y).)
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    Just sketch the graph, it should then be pretty clear whether the function is 1:1 or not.
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    A function f is injective (a.k.a. "1-1") if whenever a \ne b, f(a) \ne f(b). That is, distinct points are mapped to distinct points. Equivalently, it's injective if f(a)=f(b) implies that a=b.

    So for example f(x)=x^2 isn't injective, since 1 \ne -1 but f(1)=1^2=1=(-1)^2=f(-1).

    However, f(x)=e^x is injective, since if e^a=e^b then \ln(e^a)=\ln(e^b), so a=b.

    A good way of telling if a map is injective is to draw a graph. If you held an infinitely long ruler horizontally anywhere over the graph, it should only ever intersect the graph at one point. Otherwise, it's not injective.
 
 
 
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