Examining the nature of mapping of $f(x) = \frac{x}{x^2 - 2}$.

Question

Let f: $\mathbb{Q} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{x^2 - 2}$, $x \in \mathbb{Q}$. Examine the nature of mapping

Attempt: I think f(x) is not onto. Since Irrational Numbers which are subset of $\mathbb{R}$ are being unmapped. Also it seems one-one to me, But I'm not sure. Can anybody explain the nature of this mapping.

Answer 1

$$x\in \mathbb{Q} \Rightarrow x^2-2\in \mathbb{Q} \Rightarrow \frac{x}{x^2-2}\in \mathbb{Q}.$$ So actually $f$ is $\mathbb{Q}\rightarrow\mathbb{Q}$, hence it is not on to. $$f(x)=f(y) \Longleftrightarrow x(y^2-2)=y(x^2-2) \Longleftrightarrow (y-x)(xy+2)=0.$$ So if $xy=-2$ and $x\neq y$, we also have $f(x)=f(y)$, hence it is not one-one.