Solution to 1. Is the function 𝑓: ℤ → ℤ 𝑓(𝑥) = 𝑥 2 + 3 injective, … - Sikademy
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Archangel Macsika

1. Is the function 𝑓: ℤ → ℤ 𝑓(𝑥) = 𝑥 2 + 3 injective, surjective or bijective? Prove your assertions

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Consider the function f:\Z\to\Z,\ f(x)=x^2+3. Since for x_1=-1 and x_2=1\ne x_1 we have that f(x_1)=(-1)^2+3=4=1^2+3=f(x_2), we conclude that the function f is not injective. Taking into account that for y=0\in\Z the equation x^2+3=0 has no integer roots, we conclude that f^{-1}(0)=\emptyset, and hence the function f is not surjective. Therefore, the function f is not bijective.

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Question ID: mtid-5-stid-8-sqid-2666-qpid-1136