Relation of bijective functions and even functions?
Solution 1:
No even function from $\mathbb R$ into $\mathbb R$ is bijective because, for instance, $f(1)=f(-1)$. Almost the same argument shows that, if $a>0$, no even function from $(-a,a)$ into $\mathbb R$ is bijective (or, indeed, injective).
Note that some odd functions from $\mathbb R$ into $\mathbb R$ are not bijective too. Take, say, $f(x)=x^3-x$.