Given $f :\{1,2 ,4 \} \rightarrow \{a,b,c,d\} $ and $g : \{a,b,c\} \rightarrow \{1, 2, 4\}$ where $f = \{(1,b),(2,d),(4,d)\}$ and $g = \{(a,1), (b,2), (c,4)\}$.
What is $g \circ f$?
As far as I know the image of $f$ needs to be subset of the domain of $g$, and in this case it isn't. So what is then $g \circ f$, is it an empty set or can't the computation be even obtained? Thanks.
On
The two functions $f$ and $g$ cannot be composed as $g\circ f$ since $g\circ f(2)$ and $g\circ f(4)$ are not defined.
So we say the composed function $g\circ f$ $\color{red}{\text{does not exist}}$ for these two functions $f$ and $g$.
However, you can define the restricted function $g\circ f\Big|_{\{ 1\}}=\{ (1,2)\}$ by restricting the domain of $f$ to $\{ 1\}$.
$g \circ f $ is a function from $\{1,2,4\} $ to $\{1,2,4\} $.
Let us be explicit $$g\circ f (1)=g (f(1))=g (b)=2$$
$$g\circ f (2)=g (f (2))=g (d)$$
but $g (d )$ is not defined.
Thus $g\circ f (2) $ does not exist.
You can finish.
What is the domain of $g\circ f $.