Composition of functions

Definition

Composition of functions is the process of applying one function to the results of another. If $f(x)$ and $g(x)$ are two functions, the composition is written as $(f \circ g)(x) = f(g(x))$.

5 Must Know Facts For Your Next Test

1. The order in which functions are composed matters: $(f \circ g)(x)$ is generally not equal to $(g \circ f)(x)$.
2. To find the domain of a composite function, consider the domain of the inner function and ensure it fits within the domain of the outer function.
3. Composition can be used to simplify complex expressions by breaking them down into simpler functions.
4. Composite functions can be evaluated at specific points by substituting values step-by-step through each function.
5. If $f(x)$ and $g(x)$ are inverses, then $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$.

Review Questions

• What is the composition of $f(x) = 2x + 3$ and $g(x) = x^2$?
• How do you determine if two functions are inverses using composition?
• What is the domain of $(f \circ g)(x)$ if $f(x) = \sqrt{x}$ and $g(x) = x - 4$?

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, such that if $f(g(x)) = x$ and $g(f(x)) = x$, then $g$ is the inverse of $f$.

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