ON INVERSE FUNCTIONS
Composition of Functions
Suppose the rule of function f(x) is 
and the rule of function g(x) is 
. Suppose now that you want to "leapfrog" the functions as follows: Take a 2 in the domain of f and link it to 9 with the f(x) rule, and then take the 9 and link it to 157 with the g(x) rule. This is a lot of work and you would rather just work with one function, a function that would link the 2 directly to the 157.
Since the g function operates on f(x), we can write the composition as g(f(x)). Let’s call the new function h(x) = g(f(x)). You can simplify from the inside out or the outside in.
Inside Out: 
Let’s check to see if the above function will link 2 directly to 157.
It does.
Outside In: 
You can see that it is the same as the function we derived from the inside out.
The following is an example of finding the composition of two functions.
Example 3: Find 
and 
if 
and 
Solution - :
Inside Out: 
Outside In: 
Check:
The only values of x that work for g are those values of x such that 
. The domain of f(g(x)) is the set of real numbers such that 
. Now for what values of x is 
? 
when 
. Therefore, if you reset your display boundaries on your graphing calculator so that you are to the left of x = -58, you will see the graph. The domain of the composite function f(g(x)) is the set of real numbers in the interval 
.
Solution - 
:
Check:
.
Let’s see if the new function, let’s call it h(x), will link the 9 directly to .
Comment:
What is the domain of the composite function g(f(x))? Is it the same as the domain of the composition function f(g(x))?
The domain of the f(x) is the set of real numbers in the interval 
, and the domain of the composite function is the set of real numbers in the interval 
. Which one is it? The domain of the composite function is the set common to both, or 
.
Choose any number in the domain to check the function. Let’s try x = 40. The function f links 40 to 
, and the function g links to 0.585786…
Let’s see if the new function will link the 40 directly to 0.585786…
Review another example of finding the composition of functions.
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Author:Nancy Marcus