Domain of a composite function

Definition

The domain of a composite function $f(g(x))$ is the set of all real numbers $x$ such that $x$ lies in the domain of $g(x)$ and $g(x)$ lies in the domain of $f(x)$. It ensures that both functions are defined for the input values used.

5 Must Know Facts For Your Next Test

1. To find the domain of a composite function, first determine the domains of the individual functions.
2. The composite function $f(g(x))$ is only defined if $x \in \text{domain}(g)$ and $g(x) \in \text{domain}(f)$.
3. If either condition fails, then those values must be excluded from the domain of the composite function.
4. Graphical analysis can help visualize where both functions overlap to determine their joint domain.
5. Using interval notation can clearly express the resulting domain after compositing two functions.

Review Questions

• How do you determine if a value is in the domain of a composite function?
• What are the steps to find the domain of $f(g(x))$ if given specific functions for $f$ and $g$?
• Explain why it's necessary to check both domains when forming a composite function.

Related terms

Domain: The set of all possible input values (usually x-values) for which a function is defined.

Range: The set of all possible output values (usually y-values) produced by a function.

Function Composition: The process of applying one function to the results of another, denoted as $(f \circ g)(x) = f(g(x))$.