Combining functions

Definition

Combining functions involves creating a new function by applying one function to the results of another. This process is also known as the composition of functions.

5 Must Know Facts For Your Next Test

1. The notation for the composition of two functions $f$ and $g$ is $(f \circ g)(x) = f(g(x))$.
2. To compute $(f \circ g)(x)$, first apply $g$ to $x$, then apply $f$ to the result of $g(x)$.
3. The domain of $(f \circ g)(x)$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.
4. Composition of functions is not generally commutative; $(f \circ g)(x) \neq (g \circ f)(x)$ in most cases.
5. When decomposing a composite function, identify inner and outer functions, where the outer function is applied last.

Review Questions

• What does the notation $(f \circ g)(x)$ represent?
• How do you find the domain of a composite function?
• Is composition of functions commutative? Provide an example.

Related terms

Function:

A relation between a set of inputs and a set of permissible outputs where each input is related to exactly one output.

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.