<s>[INST] Is $f(x)=\frac{x^2}{1+x^2}$ bijective? $f: \mathbb{R} \to \mathbb{R}$ via $f(x) = \frac {x^2}{1+x^2}$
Injective:
Suppose f(a) = f(b)
$$ \implies \frac {a^2}{1+a^2} = \frac {b^2}{1+b^2} \implies a^2 +a^2b^2 = b^2 +b^2a^2$$
$$ \implies a^2 = b^2 \implies a \neq b$$
Surjective: If $x<0$, the image of $f$ is $[0, \infty)$
Hence $f$ is not surjective.
Therefore, $f$ is neither injective or surjective.
 [/INST] $f $ is an even function
$$\implies f (1)=f (-1)=\frac {1}{2} $$
$\implies f $ is not injective
$\implies f $ is not bijective.
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