Intermediate Math Solutions – Functions Calculator, Function Composition

Function composition is when you apply one function to the results of another function. When referring to applying g(x) to f(x), the function composition is denoted as (f\:\circ\:g)(x),\:or\:f(g(x)).

How does function composition work?

x is only used as a place holder in the function. We can substitute in other values for x. When given the function composition (f\:\circ\:g)(x), we take whatever g(x) is equal to and input it in for x in f(x). This will give us the function composition. It is important to know that (f\:\circ\:g)(x)\ne(g\:\circ\:f)(x).

Make sure that the domain of the inner, first function is respected in the function composition.

Let’s go through an example step by step to help you better understand how to solve function compositions.

Here’s an example (click here):

Given f(x)=2x+3 and g(x)=-x^2+5,find g(f(x+3)).
* g(f(x+3)=g(x)\circ(f(x+3))

1. Substitute x+3 for x in f(x)

2. Substitute f(x+3) for x in g(x)

g(2x+9)

3. Evaluate g(2x+9)

Here’s another example (click here):

Last example (click here):

Function composition is very simple. My only piece of advice is to double check your work! It is so easy to have a simple calculation error that will cost you a point or two. Try your best to avoid this and double check your work. Practice a few tricky problems and you’ll be all set to move on from this topic.

Until next time,

Leah