1. **Problem statement:** We are given a function $\varphi : \langle \mathbb{Z}, + \rangle \to \langle \mathbb{Z}, + \rangle$ defined by $\varphi(x) = x^2$. We need to determine if $\varphi$ is a group homomorphism. 2. **Recall the definition of a group homomorphism:** A function $\varphi$ between two groups $(G, +)$ and $(H, +)$ is a homomorphism if for all $a, b \in G$, the following holds: $$\varphi(a + b) = \varphi(a) + \varphi(b)$$ 3. **Apply the definition to $\varphi(x) = x^2$:** We check if $$\varphi(a + b) = (a + b)^2 = a^2 + 2ab + b^2$$ equals $$\varphi(a) + \varphi(b) = a^2 + b^2$$ 4. **Compare both sides:** $$a^2 + 2ab + b^2 \neq a^2 + b^2$$ unless $2ab = 0$ for all integers $a, b$, which is not true in general. 5. **Conclusion:** Since $\varphi(a + b) \neq \varphi(a) + \varphi(b)$ in general, $\varphi$ is not a group homomorphism. **Final answer:** $\varphi$ is not a homomorphism.