Let $f:A\rightarrow\mathbb{R},g:B\rightarrow\mathbb{R},g \circ f:C\rightarrow\mathbb{R}$
Then we have:
1.$C\subseteq A$
2.$f(C)\subseteq B$
Proof.
$\forall c \in C,g \circ f$ is defined on c
$\Rightarrow f$ is defined on c that $ c \in A$
and $g$ is defined on $f(c)$ that$ f(c)\in B$
$\Rightarrow \forall c \in C,c \in A$ and $f(c)\in B$
$\Rightarrow C\subseteq A$ and $f(C)\subseteq B$
Questions:
1.Are these statements true?
2.Does my proof look fine?
3.Is there any other nice relationship between
domain,codomain and range
with condition $f:A\rightarrow\mathbb{R},g:B\rightarrow\mathbb{R},g \circ f:C\rightarrow\mathbb{R}$?
Thanks :)
Definitions I'm using:
$f:A\rightarrow{B}$:
$f$:domain $\rightarrow$ co-domain
domain:
Subset of $\mathbb{R}$ that $f$ is defined on
(for example, domain of $\frac{1}{x}$ is $\mathbb{R}\setminus\{0\}$)
co-domain:
$\mathbb{R}$ as default
range:
Outputs of $f$ as a subset in co-domain