Composite Functions

Composite functions are a topic covered in pre-calculus, but they come back in calculus in the form of the Chain Rule, so here is a small review.

A composite function is a function that takes of the output of one or more functions as input. Essentially, a composite function is any function that “calls” another function to modify its input, then messes with whatever that other function returns.

Take the function $h(x)=\cos (x)+1$ as an example. $h(x)$ calls on $\cos (x)$ to modify its input $x$ value, then adds one to it. That means that $\cos (x)$ is one of the elements of the composite function, since it is a function of its own. Let’s give it a different name— $f(x)=\cos (x)$. So, $h(x)$ feeds its own $x$ into $f(x)$. $h(x)=f(x)+1$; $h(f(x))=\cos (x)+1$. You’re still getting the same result.

$f(x)$ is not the only other function that we can say makes up $h(x)$. $h(x)$ also adds one to the output of $f(x)$. This could be defined as another function, $g(x)=x+1$. $g(x)$ takes its input and adds one to it. $h(x)$, then, could be described in terms of two other functions: $h(x)=g(f(x))$.

Consider a function in the terms of programming.

int f(x) {
	return cos(x);
}
 
int g(x) {
	return x+1;
}
 
int composite = g(f(x));

Composite relies on both the outputs of g(x) and f(x), which both modify x in some way.