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)). 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 1:** Find and if and .

**Solution - ****:**

**Inside Out:**

The composition function is equal to

__Outside In:__

**Check:**

Let’s see if the new function, let’s 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:**

The composition function is equal to

**Outside In:**

**Check:**

Let’s see if the new function, let’s call it h(x), will link the 2 directly to the 9.

It does and the new composition function can be written

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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