A function $f:X\rightarrow Y$ maps each $x\in X$ to some $y \in Y$. So consider $\tan{\frac{\pi}{2}}$ for which $\tan(x)$ is undefined, so in this case, $\tan(x)$ does not map to an element of its range. This conflicts with my understanding of what a function is. Why do we still consider $\tan(x)$ a function?
[Math] Why is $\tan(x)$ a function
functionstrigonometry
Best Answer
The set $X$ in your definition is the domain of the function. The domain of $\tan(x)$ is typically taken to be
$$ X=\bigcup_{k\in\Bbb{Z}} \left(-\frac{\pi}{2}+k\pi,\frac{\pi}{2}+k\pi\right) $$
Thus $\pi/2\notin X$, and so don't need to assign a value to $\tan(\pi/2)$ (or for any $\pi/2+k\pi,k\in\Bbb{Z}$ for that matter).