Composite Functions

So you may already know about basic functions such as linear or quadratic (which we also cover here), but perhaps you have not heard the term composite functions. A composite function is a function that is plugged into another function. If you have already looked into these types of functions before, you may have seen one or two types of notation regarding it. Notation 1:

f(g(x))

This is perhaps the clearest way of putting it. It shows clearly how the function g is plugged into the function f. These functions can of course contain any name as mentioned in linear and quadratic functions. Notation 2:

(f ∘ g)(x)

Perhaps this looks more scary, but the concept is the exact same as in notation 1. Just a different notation. Let's look at how they work. If I give you the functions:

f(x) = 2x

g(x) = 3x +4 

Then we say that the composite function f(g(x)) = 2 × (3x+4) + 4. You may simplify this further, but this is everything you need to know about composite functions in a nutshell. 

So what did we do? If, you take a closer look, what we did was just replace all the x's in function f, with function g. Where function f originally had an x, it now has a 3x+4 which is the g function

Given f(x) = 3x^2 and g(x) = 4x + 5 and h(x) = 2x, find (f ∘ g ∘ h)(x).

Substitute plug the function g into function f. So all x will be replaced with the function of g. Then plug function h into g  by substituting each x in it by function h. So it will be:

(f ∘ g ∘ h)(x) = 3 × (4 × (2x) + 5)^2

You may simplify this, but this is all in a nutshell.