A car body is pretty useless without an engine.
An engine is almost as useless when not in an engine compartment.
But put an engine in a car and you have a useful and versatile means of transportation.
Most useful functions in mathematics are built up from simpler functions. Functions whose outputs are numbers can be combined by addition, subtraction, multiplication, and division.
When the inputs of the functions are also numbers, another common way to combine them is to use the output of one function as the input of another.
This is called the composition of functions. The composition of two functions f and g is written like this, and is read “f of g.”
The definition of f of g is: f of g of x is equal to f of g of x. This sounds better with a brief pause: f of g, of x, is equal to, f of, g of x.
This means the output of g is being used as the input for f. g of x is computed first, and result is “plugged in” to f.
This simulation will allow you to view the graphs of the compositions of pairs of several important functions.
Choose a function f of x by selecting one of the nine buttons in the top row of buttons, …
… then choose a second function g of x by selecting one of the nine buttons in the second row.
You can get a look at what the individual functions look like before composing them, if you wish, by using the checkboxes on the left.
When you have selected the functions whose composition you want to graph, choose the particular composition in which you are interested from the radio buttons to the right.
You can view the graph of neither, either, or both of the uncomposed functions at the same time as the graph of the composition, and you can switch to the other composition at any time.
You can even compose a function with itself by selecting the same numbered functions from the rows for f and g.
Try all possible compositions of functions. Some compositions are more interesting than others, and some convey important mathematical truths. Good luck, and keep your composure! 1