Question

Show that the function $f:R\to R$, defined by $f\left(x\right)=x^2-\frac{x^2}{1+x^2},$ is neither one-one nor onto.

Answer 1

$f\left(x\right)=\frac{x^4}{1+x^2}$
$f(-x)=\frac{(-x{)}^4}{1+(-x{)}^2}=f\left(x\right)$
$\Rightarrow f\left(x\right)$ is an even function, hence it is many one.

Since, $f\left(x\right)\ge 0\forall x\in R$, so $f\left(x\right)$ does not take any negative value.
So, range of $f\left(x\right)$ is not $R$.
Hence, it is not onto.
$\Rightarrow f\left(x\right)$ is neither one-one nor onto (proved).