Functions, like numbers, are a system of things (functions) and rules (operations).

• the sum of two functions is another function.
• the difference of two functions is another function.
• the product of two functions is another function.
• the quotient of two functions is another function.

Of course, we always have to pay some attention to domains.

The usual attention to domains includes adopting the intersection of domains to be the new domain, and avoiding domain numbers which make the denominator equal to $0$.

In addition to these four familiar operations, functions have another operation: composition.

We have already peeked at this. We took a single output number of one function and used it as the input to anther function. We can extend this idea to the whole function.

The composition of F and G, symbolized by $F \circ G$ or $F(G)$, is a new function. Its domain comes from the domain of $G$. Its range comes from the range of $F$.

In this section, we will restrict our study to the composition of linear functions.

Learning Outcomes

In this section, students will

• identify implied domains.
• create formulas for compositions.
• create graphs for compositions.