# Composition of Functions

The function whose value at $x$ is $f\left(g\left(x\right)\right)$ is called the composite of the functions $f$ and $g$ .  The operation that combines $f$ and $g$ to produce the composite is called composition .

Notation:  $\left(f\circ g\right)\left(x\right)$ or $f\left(g\left(x\right)\right)$

The domain of $f\left(g\left(x\right)\right)$ is the set of all $x$ in the domain of $g$ such that $g\left(x\right)$ is in the domain of $f$ .

Example:

If $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=3x-5$ , then find $f\left(g\left(4\right)\right)$ .

Substituting $4$ for $x$ in the definition of the function $g$ , $g\left(4\right)=3\left(4\right)-5=7$ . Then, $\text{\hspace{0.17em}}f\left(g\left(4\right)\right)=f\left(7\right)$

$f\left(7\right)=\sqrt{7}$