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    Hi, I have a test tomorrow on calculus and was hoping someone could give me a hand.

    I would appreciate it if someone could explain to me the best method for proving a function is Surjective and Injective. I have solutions, but they don't make much sense to me

    E.g. given that f and g are injective functions, prove that f(g(x)) is also injective.

    Thanks
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    f is injective if f(x) = f(y) implies that x=y. Equivalently, if x \ne y then f(x) \ne f(y). That is, a function is injective if it maps distinct points to distinct points, so that a point is determined by its image (i.e. there is a well-defined inverse on a subset of the codomain).

    It's usually easier to work with the definition that f is injective if f(x) = f(y) \Rightarrow x=y.

    So if f and g are injective and f(g(x)) = f(g(y)), then...?

    A function is surjective if every element of the codomain can be written as f(x) for some x. For example, consider f(x)=x^2 on the real numbers. This isn't surjective, since you can't write -1 as x^2 for any real value of x. But the function f(x)=x^3 is surjective, since if y is any real number and x=\sqrt[3]{y} then f(x)=y, so we can write every real number in the form f(x).
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    (Original post by nuodai)
    f is injective if f(x) = f(y) implies that x=y. Equivalently, if x \ne y then f(x) \ne f(y). That is, a function is injective if it maps distinct points to distinct points, so that a point is determined by its image (i.e. there is a well-defined inverse on a subset of the codomain).

    It's usually easier to work with the definition that f is injective if f(x) = f(y) \Rightarrow x=y.

    So if f and g are injective and f(g(x)) = f(g(y)), then...?

    A function is surjective if every element of the codomain can be written as f(x) for some x. For example, consider f(x)=x^2 on the real numbers. This isn't surjective, since you can't write -1 as x^2 for any real value of x. But the function f(x)=x^3 is surjective, since if y is any real number and x=\sqrt[3]{y} then f(x)=y, so we can write every real number in the form f(x).
    Thank You.
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    You might also want to have a look at http://gowers.wordpress.com/2011/10/...-and-all-that/
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    (Original post by DFranklin)
    You might also want to have a look at http://gowers.wordpress.com/2011/10/...-and-all-that/
    I admit to having been somewhat confused at first by the start of that blog, regarding the use of "range" to mean "codomain". The commentaries at the end are most enlightening though.
 
 
 
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