Function Notation f(x) and Functions of Functions f(g(x))

We take a look at creating compositions of functions. How do you handle situations in which you have a function of a function? (Or a function in a function.) Our examples include the function that adds three to every input, f(x) = x + 3, and the function that quadruples every input, g(x) = 4x. We create compositions with these functions, including f(f(x)), which is the function that adds three and then adds three again. We also examine f(g(x)), which means to take an input, apply the function g to it, and use the output of g as the input for the function f. In other words, f(g(x)), would be the function that quadruples an input and then adds 3 to the result. This arrangement of function is sometimes referred to as nested functions.