key term - Domain of a composite function

Definition

The domain of a composite function is the set of all input values for which the composed function is defined. It must satisfy the domains of both the inner and outer functions.

5 Must Know Facts For Your Next Test

1. To find the domain of a composite function $f(g(x))$, first determine the domain of $g(x)$, then check where $f$ is defined using these outputs from $g$.
2. If any value in the range of $g(x)$ is not in the domain of $f$, then that value must be excluded from the domain of $f(g(x))$.
3. The notation for a composite function is generally written as $(f \circ g)(x)$ or $f(g(x))$.
4. A common mistake is to only consider the domain restrictions of one function; both must be checked.
5. Graphically, you can find the domain by ensuring that points on the graph of $g$ map to points within the valid range for $f$.

Review Questions

Related terms

Function Composition:

Combining two functions where the output of one function becomes the input for another.

Domain:

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

Range:

$\text{The set}$ $\text {of all possible output values produced by a given function}.$