Composite Functions and Almonds f(g(x))

Here's a tough concept for high school math students, composite functions (for more on functions click here)

Definition of Composition of Functions

• Composition of Functions is the process of combining two functions where one function is performed first and the result of which is substituted in place of each x in the other function. 

Yet, what does this actually mean? And how can a young student of math make sense out of such abstract definitions. Let's call upon an analogy using almonds.


Let our ultimate goal be to have roasted almonds mixed with other assorted nuts. Thus, we have two machines; one that roasts the almonds and one that adds other assorted nuts in with the roasted almonds.

But, loosely speaking, a machine is simply a function; it transfers inputs to outputs (or in the language of math it transfers domains to ranges)

Let us revisit our example

(Here is a picture of our process f(g(x))

(Let x, our input, be regular almonds)

(We feed our regular almonds in our first machine, call it our g machine, which roasts our almonds)

(Our g machine takes plain almonds and roasts them, thus, it takes almonds x and roasts them making them g(x). We then take our roasted almonds and use them as our input in the next machine) 

(Here is out next machine, our f machine. Our f machine takes roasted almonds and adds assorted nuts )

(Thus, our final output for our regular almonds after traveling through two machines is roasted almonds with assorted nuts; f(g(x))

(Here is how such a concept would look in a textbook)