Answer to Question #241807 in Discrete Mathematics for vinh hoang

Question

determine whether these functions are bijections f:Q to R x²+1/x

Answer 1

Let us determine whether the function $f:\mathbb Q \to \R,\ f(x)= x^2+\frac{1}x,$ is a bijection.

Let $x\in\mathbb Q,$ that is $x$ be a rational number. Taking into account that the sum, the product and the fraction of two ratianal number is a rational number, we conclude that $x^2\in\mathbb Q,\ \frac{1}x\in\mathbb Q,$ and hence $f(x)=x^2+\frac{1}x\in\mathbb Q.$ Therefore, for any $y\in\R\setminus\mathbb Q\subset\R$ we get that the preimage $f^{-1}(y)$ is emptyset, and thus $f$ is not a surjection. Consequently, $f$ is not a bijection.