Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 months ago

Determine whether the function is one-to-one or onto?

Determine whether the following function is one-to-one or onto, justifying 

with proof or counterexample

g : Z → Z given by g(n) =  2n + 2,  if n even   or   n, if n odd

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  • Pope
    Lv 7
    2 months ago

    Let p and q be integers such that g(p) = g(q).

    Suppose p and q both even.

    2p + 2 = 2q + 2

    2p = 2q

    p = q

    Suppose p and q both odd.

    p = q

    Suppose p even and q odd.

    2p + 2 = q

    q = 2(p + 1)

    That would make q an even number, which contradicts the hypothesis.

    Therefore g(p) = g(q) only if p = q.

    Function g is one-to-one.

    Zero is an integer, but there exists no integer r such that g(r) = 0.

    Function g is not onto.

  • 2 months ago

    A function is not a one-to-one function if at least two points of the domain are ... Proof: Suppose x1 and x2 are real numbers such that f(x1) = f(x2). ... On the other hand, to prove a function that is not one-to-one, a counter example has to be given. ... g(n) = 2n - 1 for all n Z. Prove that g is not onto by giving a counter example.

  • Ian H
    Lv 7
    2 months ago

    This link explains the concepts clearly with diagrams and examples.

    https://www.aplustopper.com/one-to-one-and-onto-fu...

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