key term - Decompose a composite function

Definition

To decompose a composite function means to break it down into two or more simpler functions whose composition results in the original function. It helps in understanding the underlying structure and behavior of complex functions.

5 Must Know Facts For Your Next Test

1. Decomposition involves identifying inner and outer functions, typically denoted as $g(x)$ and $f(g(x))$ respectively.
2. Given a composite function $h(x) = f(g(x))$, $g(x)$ is applied first followed by $f$.
3. Decomposing can simplify solving equations, finding derivatives, and integrating complex functions.
4. Decomposition is not always unique; multiple valid decompositions can exist for a given composite function.
5. It is crucial to verify that the domain of the composed function matches the domains of individual functions involved.

Review Questions

Related terms

Composite Function:

A function formed by applying one function to the result of another, denoted as $(f \circ g)(x) = f(g(x))$.

Function Composition:

The process of combining two functions where the output of one becomes the input for another.

$Domain$: $The set of all possible input values for which a function is defined.$