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 rule 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)). Letís call the new function h(x) = g(f(x)). You can simplify from the inside out or the outside in.

Inside Out:

Letís 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 1: Find and if and .

Solution - :

Inside Out:

The composition function is equal to

Outside In:

Check:

The function g(x) links a 5 to the number 25 because and the function f(x) links the 25 to the number 668 because .

Letís see if the new function, letís call it h(x), will link the 5 directly to the 668.

It does and the new composition function can be written

Solution - :

Inside Out:

eqnarray4

The composition function is equal to tex2html_wrap_inline13

Outside In:

eqnarray6

Check:

The function f(x) links 2 to the number 1 because tex2html_wrap_inline15 and the function g(x) links the 1 to the number 9 because tex2html_wrap_inline17.

Letís see if the new function, letís call it h(x), will link the 2 directly to the 9.

tex2html_wrap_inline19

It does and the new composition function can be written

displaymath11

Students often ask why are these functions different. They assume that the new function created by f(g(x)) is the same function created by g(f(x)).

Wrong Assumption! Sometimes they are equal, but most of the time, these functions are different.

 

Review another example of finding the composition of functions.

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Author:Nancy Marcus

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