The order of composition matters: $f(g(x))$ is generally not the same as $g(f(x))$.
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 $f$.
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.
Composition can be used to simplify complex expressions by breaking them into simpler parts.
$(f \circ g)(x)$ 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 $g$?
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 ${f}^{-1}(y)$.