We have operations for our numbers. Operations usually take two numbers and exchange them for a third number.

Just like numbers, functions have an arithmetic. They have all of the usual number operations, plus one more. Functions also have an operation called composition. Composition takes two functions and exchanges them for a third function.

Just like number operations have an identity number, $0$ and $1$, composition has an identity function - the identity function: $Id(x) = x$.

The identity for an operation is an object which appears to be unaffected by the operation.

• Adding $0$ with numbers appears to do nothing.
• Multiplying $1$ with numbers appears to do nothing.
• Composing $Id$ with functions appears to do nothing.

We call this symmetry.

The identity is extremely important for each operation. The identity can be viewed as the center of the operational structure. We like to know how move from it and move toward it without operation. Pairs of items which produce the identity are called inverses.

• If $a + b = 0$, then $a$ and $b$ are called additive inverses.
• If $a \cdot b = 1$, then $a$ and $b$ are called multiplicative inverses.
• If $a \circ b = Id$, then $a$ and $b$ are called compositional inverses.

Given a function, we would like to know its compositional inverse, which we just call the inverse function.

Learning Outcomes

In this section, students will

• explore function arithmetic.
• investigate inverse functions.