function composition

in mathematics, function composition

below is an example of a composite function, defined with many statements together
$f(a)=\{\begin{array}{ll}x+1& \text{if }a<2\\ 2& \text{if }a=5\\ 3& \text{if }a>7\end{array}$

composite functions that join two functions together use a symbol $\circ$ to indicate which functions make up the composite function
for example, the statement $h=g\circ f$ means that $h(x)=g(f(x))$
notice that function at the front takes the output of the other function as its input

if a composite function $h$ joins two exact same functions called $f$, $h$ can be depicted as $h(x)=(f\circ f)(x)=f(f(x))=f^2(x)$