Refresher - Definition of Composition of Functions

Refresher

It is possible to combine two functions by adding, subtracting, multiplying or dividing two given functions.
There is another way to combine two functions to create a new function. It is called composition of two functions. It is a process through which we will substitute an entire function into another function. An important skill to have in this lesson is evaluation of functions.
First let's get acquainted with the notation that is used for composition of functions. When we want to find the composition of two functions we use the notation (${\displaystyle f\circ g)\left(x\right)}$ . ${\displaystyle \left(f\circ g\right)\left(x\right)}$ is defined to be ${\displaystyle f(g\left(x\right)}$).

Definition:

If f and g are two functions, the composition f and g, written (${\displaystyle f\circ g}$), is defined by the equation

Provided that ${\displaystyle g\left(x\right)}$ is in the domain of ${\displaystyle f}$. The composition of ${\displaystyle g}$ and ${\displaystyle f}$, written ${\displaystyle \left(g\circ f\right)}$ is defined by

Provided that ${\displaystyle f\left(x\right)}$ is in the domain of ${\displaystyle g}$.

Note that ${\displaystyle f\circ g}$ is not the same as ${\displaystyle f\bullet g}$, the product of ${\displaystyle f}$ and ${\displaystyle g}$.
Recall our notation for evaluating a function. If we are given a function ${\displaystyle f\left(x\right)}$and are asked to find ${\displaystyle f\left(3\right)}$, we would go to the f function, and everywhere there was an ${\displaystyle x}$, we would replace it with a ${\displaystyle 3}$.

So if our notation is now ${\displaystyle f\left(g\left(x\right)\right)}$., that says go to the f function and everywhere there is an ${\displaystyle x}$, replace it with the function ${\displaystyle g\left(x\right)}$.