Math Insight

Function composition

Video introduction

The composition $f \circ g$ of two functions $f$ and $g$ is the function formed by first applying the function $g$ and then the function $f$. The following video gives a introduction into the concept of function composition.

Summary

To apply the composition $f \circ g$ to an input $x$, you perform the following two steps. You first apply the function $g$ to the input $x$ and obtain the result $g(x)$ as the output. Next, you apply the function $f$ using $g(x)$ as the input and obtain the result $f(g(x))$ as the output. We can write the composition as $(f \circ g)(x) = f(g(x))$.

One can illustrate function composition using the function machine metaphor by connecting function machines together.

You can see some examples of composing functions.