Answer to Question #234255 in Discrete Mathematics for mickey

Question

Determine whether each of these functions is a bijection from R to R?f (x) = (x2 + 1)/(x2 + 2)

Answer 1

Let us determine whether the function $f:\R \to \R,\ f (x) = \frac{x^2 + 1}{x^2 + 2},$ is a bijection. Taking into account that for $x_1=-1$ and $x_2=1\ne x_1$ we get

$f(x_1)=f(-1)= \frac{(-1)^2 + 1}{(-1)^2 + 2}=\frac{2}{3}=\frac{1^2 + 1}{1^2 + 2}=f(1)=f(x_2),$

we conclude that the function $f$ is not an injection, and hence this function is not a bijection.