Composite function

Definition

A composite function is a function created by applying one function to the results of another. It is denoted as $(f \circ g)(x) = f(g(x))$.

5 Must Know Facts For Your Next Test

1. The notation $(f \circ g)(x)$ means $f(g(x))$, where $g$ is applied first and then $f$.
2. To find the domain of a composite function, ensure that the outputs of $g(x)$ are within the domain of $f$.
3. Composite functions are not necessarily commutative; $f(g(x))$ is generally different from $g(f(x))$.
4. If both $f$ and $g$ are continuous, then their composite function is also continuous.
5. When decomposing a composite function, identify inner and outer functions: for example, in $h(x) = (2x+3)^2$, let $u=2x+3$, then $h(u)=u^2$.

Review Questions

• What does the notation $(f \circ g)(x)$ represent?
• How do you determine the domain of a composite function?
• Why are composite functions not necessarily commutative?

Related terms

Function:

A relation between a set of inputs and a set of permissible outputs, typically represented by $f(x)$.

Domain:

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

Range:

The set of all possible output values produced by a function.