8.0 Composition of a Function

8.1 Properties of Composition of a Function
8.2 Composition of a Real Function

Let $f:A \to B$ and $g:B \to C$ be two functions. Then a function $gof:A \to C$ defined by, $$(gof)(x) = g(f(x)),\;\;\forall \;x \in A$$ is called the composition of $f$ and $g$.

Explaination:

Let $A$ , $B$ and $C$ be three non-void sets and let $f:A \to B$ and $g:B \to C$ be two functions. Since $f$ is a function from $A$ to $B$, for each $x \in A$ there exists a unique element $f(x) \in B$.

Now, as $g$ is a function from $B$ to $C$, for each $f(x) \in B$, there exists a unique element $g(f(x)) \in C$. Therefore, it is clear that for every $x \in A$ there exists an element $g(f(x)) \in C$.

Note: For the composition $gof$ to exists, the range of function $f$ should be a subset of domain of the function $g$.