1.4- Compositions of Functions

The composition of two functions is when one function is applied to another. This is another way to combine the functions.

Finding the composition of the function  f with the function  g :

 g(x)  is in the domain of f, making it an input.

 f of g of x

Find  f(g(x)) when  f(x) = 2x + 5, and  g(x) = x - 6

The first step: Understanding what the problem is asking is important when solving. Instead of plugging in x for  f(x) , you are now using the function  g(x).  It becomes your new input.

Solving:

 

       f(g(x)) = f(x - 6)

                   = 2(x-6) + 5

                   = 2x - 12 +5

                   = 2x - 7

 

 Evaluating f(g(x)) when x has a specific value such as 1 or 2 is simple now that we already found f(g(x)).

When x = 1,  you simply plug that number in 2x - 7, which gives you an answer of -5.

g of f of x

g(f(x)) consists of the same methods of solving as f(g(x)), because  f and g are still both just functions.

Find g(f(x)) when  f(x) = 2x + 5, and  g(x) = x - 6

       g(f(x)) = g(2x + 5)

                  = 2x + 5 - 6

                  = 2x - 1

 

As you can see, g(f(x)) ≠ f(g(x))  in this case, but sometimes they are equivalent.

Video:  http://www.youtube.com/watch?v=S4AEZElTPDo (This video explains all the concepts I went over about the composition of two functions clearly.)

Identifying Functions 

As well as the composition of functions, there is also identifying functions. The function h is an example used to identify two functions.

For example if h = (2x-7)³, then for f(g(x)),  f = x³ and g = 2x-7

Finding the inner and outer functions is the whole concept of identifying them. f is the outer while g is the inner.