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# 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

### 3 Answers

- PopeLv 71 month 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.

- 1 month 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.