Inverse image of an element in co-domain but not in range?

Question

Sorry, quite new to this.

I have a question that contains the image below of $g:X\rightarrow Y$ and it is asking for the inverse image of $u$. Am I correct in thinking that the answer is $\emptyset$? Or is it undefined?

Thanks in advance.

Answer 1

The inverse image of a set $S$ under $f:X\rightarrow Y$ is defined as $$f^{-1}[S]=\{x\in X:f(x)\in S\}$$ Generally the inverse image of an element $u$ is defined as the inverse image of the set $\{u\}$, which we can see is $\{x\in X:f(x)=u\}$. Given the diagram you posted, there are no such $x$'s, so the inverse image of $u$ is the empty set.