Composition of functions

Definition

The composition of functions is the application of 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 order of composition matters: $f(g(x))$ is generally not the same as $g(f(x))$.
2. The domain of the composite function $(f \circ g)(x)$ is determined by the domain of $g$ and the domain of $f$, considering where $g(x)$ lies within the domain of $f$.
3. To verify if two functions are inverses using composition, check if $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$ for all $x$ in their respective domains.
4. Composition can be used to simplify complex expressions by breaking them into simpler parts.
5. $(f \circ g)(x)$ can be thought of as first applying function $g$ to $x$, then applying function $f$ to the result.

Review Questions

• How do you denote the composition of two functions, $f$ and $g$?
• What must be true about the domains for $(f \circ g)(x)$ to be valid?
• If $(f \circ g)(x) \= x$, what does this imply about functions $f$ and $g$?

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: if the function $f(x)$ maps an element $a$ to an element $b$, then its inverse maps element $b$ back to element $a$. Notationally, this is written as ${f}^{-1}(y)$.

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