Composition is an operation on functions. It takes two functions and produces a third function.

$$NewFunction(d) = (Outside \circ Inside)(d) = Outside(Inside(d))$$

We have explored this idea with linear functions. Now, we would like to explore the composition of any two functions.

With just a single function, we normally think of running along the real line from left to right connecting these domain numbers with their function values. It is a very orderly process. Composition follows the same idea, except the domain of the outside function is now the range of the inside function.

Now the $Inside$ will be providing the domain numbers to the $Outside$ function and it may not be as orderly. The values of the $Inside$ function may increase and decrease, which, in turn, means the domain numbers for $Outside$ may get repeated or run backwards. The $Inside$ function takes us on a whacky ride through the domain of $Outside$, possibly repeating domain numbers and thus function values as well.

In this section, we will map the whole process.

0.1 Learning Outcomes

In this section, students will

• compose functions.