This question isn't typically associated with the level of math that I'm about to talk about, but I'm asking it because I'm also doing a separate math class where these terms are relevant. I just want to make sure I understand them because I think I may end up getting answers wrong when I'm over thinking things.
In my first level calculus class, we're now talking about critical values and monotonic functions. In one example, the prof showed us how to find the critical values of a function $$f(x)=\frac{x^2}{x-1}$$ He said we have to find the values where $f' (x)=0$ and where $f'(x)$ is undefined.$$f'(x)=\frac{x^2-2x}{(x-1)^2}$$
Clearly, $f'(x)$ is undefined at $x=1$, but he says that $x=1$ is not in the domain of $f(x)$, so therefore $x=1$ is not a critical value. Here's where my question comes in:
Isn't the "domain" of $f(x)$ $\mathbb{R}$, or $(-\infty,\infty)$? If my understanding of Domain, Codomain, and Range is correct, then wouldn't it be the "range" that excludes $x=1$?
Best Answer
If $f:X\to Y$ is a function from $X$ to $Y$, we usually call $X$ the domain, $Y$ the codomain, and $f(X)$ the range. Some authors call $Y$ the range, in which case $f(X)$ is called the image.