*theorem*I gave you in the Introductory post for this series?

To test whether these two functions are

*inverses*, we need to look at the two*composite functions*in the*theorem*above.**Step 1: Evaluate the first**

*composite function*in the*theorem*above, (f o g)(x).To form a

*composite function*, locate the

*variable (x)*in the first

*function*(circled in

**red**in the image above). Insert the

**entire second**

*function*into this

*variable*(circled in

**green**, above, with arrow showing the insertion point).

Now, evaluate this new

*function*, shown below.

**If the first**

*composite function*equals*x*, continue to Step 2.**Otherwise, the two**

*functions*are__NOT__*inverses*.**In this example, the first**

*composite function,*(f o g)(x), does equal

*x*. Therefore, we continue to Step 2.

**Step 2: Evaluate the second**

*composite function*in the*theorem*above, (g o f)(x).**Form this**

*composite function*as before. This time, the entire first

*function*(circled in

**red**above, with arrow showing insertion point) is inserted into the

*variable placeholder*of the second

*function*(circled in

**green**).

As before, evaluate this new

*function*, shown below.

**If the second**

*composite function*equals*x*, the functions__ARE__*inverses*.

**If**

__EITHER__of the two*composite functions*do not equal*x*, the functions__ARE NOT__*inverses*!**In this example, the two**

*functions*are

*inverses*of each other.

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Here's another example.

Begin with

**Step 1**as before.

The first

*composite function*cannot be reduced any further than (3x^2 - 1), as shown in the image above. Therefore, it

**does not**equal

*x*. And therefore, the functions f(x) and g(x) are

__NOT__inverses.

There is no need to proceed to

**Step 2**. According to the

*theorem*, both

*composite functions*

**must equal**

*x*for the two functions to be inverses of each other.

---------------------------------------------------------------------------------------------------------

The

*composite functions*need not be evaluated in the order given. You could evaluate the second

*composite function*first. Sometimes, one of the two is easier to reduce than the other. If you get stuck trying to reduce one of them, try the other.

In the next, and final, post of this series, we'll discover how to find the inverse of a given function. :-) I hope you'll stick around for it!