Main Concept

Sometimes it is useful to apply one function, and then apply a second function to the result. For example, suppose $f\left(x\right)\=y$ and $g\left(y\right)\=z$. First, $f$  takes the input value $x$ and outputs $y$. Then, $y$ is input into $g$ which gives a new output, $z$. This idea can be used to create a new function, called the composition of g and f.

Let $f$ and $g$ be two functions. The composition of the functions $g$ and $f$, written $g\circ f$  ("g composed with f "), is defined by

$\left(g\circ f\right)\left(x\right)\=g\left(f\left(x\right)\right)$

The domain of the function $g\circ f$  is the set of all numbers $x$ in the domain of $f$ for which $f\left(x\right)$ is in the domain of $g$.

$f\left(x\right) \=$

$g\left(x\right)\=$

$f\left(g\left(x\right)\right)\=$$$

$g\left(f\left(x\right)\right)\=$$$

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