Composite function

Definition

A composite function is a function formed 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 you apply $g$ first and then apply $f$ to the result.
2. Composite functions are not necessarily commutative, meaning $f(g(x)) \neq g(f(x))$ in general.
3. To find the domain of a composite function, ensure that the input values for the inner function $g(x)$ fall within its domain and that the resulting values from $g(x)$ fall within the domain of the outer function $f$.
4. If both component functions are continuous, then their composite is also continuous at all points where both functions are defined.
5. When dealing with inverse functions, $(f \circ f^{-1})(x) = x$ for all $x$ in the domain of $f^{-1}$ and $(f^{-1} \circ f)(x) = x$ for all $x$ in the domain of $f$.

Review Questions

• Explain why $(f \circ g)(x)$ is not generally equal to $(g \circ f)(x)$.
• How do you determine the domain of a composite function?
• What happens when you compose a function with its inverse?

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.

Inverse Function: A function that reverses another function, denoted as $f^{-1}(x)$, such that if $y = f(x)$, then $x = f^{-1}(y)$.

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