Determine if these functions are injective 
Determine if the following functions are injective.
$$f(x) = \frac{x}{1+x^2}$$
$$g(x) = \frac{x^2}{1+x^2}$$

My answer:
$f(x) = f(y)$
$$\implies \frac{x}{1+x^2} = \frac{y}{1+y^2}$$
$$\implies x+xy^2 =y+yx^2$$
$$\implies x=y$$
Hence $f(x)$ is injective
$g(x) = g(y)$
$$\implies \frac{x^2}{1+x^2}=\frac{y^2}{1+y^2}$$
$$\implies x^2+x^2y^2=y^2+y^2x^2$$
$$\implies x^2=y^2$$
$$\implies \pm x=\pm y$$
So $g(x)$ is not injective

If $x + xy^2 = y+yx^2$, then $(x - y) = yx^2 - xy^2 = xy(x-y)$.  This gives $(x-y)(xy-1) = 0$, hence either $x=y$ or $xy=1$. You can check that for example, $f(\frac 12) = f(2) = \frac 25$.
Hence, $f$ is not injective.
The answer to the second question is correct.
