Lesson 16.6: Operations

Property

A custom operation uses a symbol (like $\star$, $\#$, $\&$, $\oplus$) to represent a specific mathematical rule involving two variables. The general form is $a \circ b = \text{expression involving } a \text{ and } b$, where $\circ$ is the custom symbol.

Examples

Property

When custom operations are nested, evaluate from the innermost operation outward, following the order of operations. For nested operations like $a \star (b \star c)$, first evaluate $(b \star c)$, then apply the operation $a \star \text{result}$.

Examples

Property

Function iteration notation: $f^n(x)$ means applying function $f$ exactly $n$ times, so $f^2(x) = f(f(x))$

Function multiplication: $f(x) \cdot f(x) = [f(x)]^2$ means multiplying the output by itself

Property

The composition of functions f and g is written $f \circ g$ and is defined by

We read $f(g(x))$ as $f$ of $g$ of $x$. In composition, the output of one function is the input of a second function.

Examples